Alma Mater Studiorum Universit`a degli Studi di Bologna ... - Inaf

Alma Mater Studiorum
Università degli Studi di Bologna
Facoltá di Scienze Matematiche, Fisiche e Naturali
Dipartimento di Astronomia
DOTTORATO DI RICERCA IN ASTRONOMIA
Ciclo XXIII
MAGNETIC FIELDS AROUND RADIO GALAXIES
FROM FARADAY ROTATION MEASURE ANALYSIS
Dottoranda:
DARIA GUIDETTI
Coordinatore:
Chiar.mo Prof.
Lauro MOSCARDINI
Relatrice:
Chiar.ma Prof.ssa
Loretta GREGORINI
Co-relatori:
Chiar.mo Prof. Robert A. LAING
Dott.ssa Paola PARMA
Settore Scientifico Disciplinare: Area 02 - Scienze Fisiche
FIS/05 Astronomia e Astrofisica
Esame Finale Anno 2011

This thesis has been carried out at:
European Southern Observatory (ESO, Garching)
and
Istituto di Radioastronomia (IRA-INAF, Bologna)
as part of the Institute research activity

To Hypatia from Alexandria in Egypt, mother of the observational method
To all Warriors of the Light
And to Lorenzo, which will arrive soon and be met with lots of love

“There are two possible outcomes: if the result confirms the hypothesis, then you’ve made a
discovery. If the result is contrary to the hypothesis, then you’ve made a discovery.”
(Enrico Fermi)
Let Light and Love and Power restore the Plan on Earth.

Contents
Abstract
i
1 Physics of radio galaxies 1
1.1 Active galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Non-thermal emission mechanisms of active galaxies . . . . . . . . . . . . . . . 2
1.2.1 Synchrotron radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Polarization of synchrotron radiation . . . . . . . . . . . . . . . . . . . . 3
1.2.3 Inverse Compton emission . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Radio galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Radio-galaxy morphologies . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Morphology and expansion of lobes and tails . . . . . . . . . . . . . . . 7
1.3.3 Internal physical conditions of radio galaxies . . . . . . . . . . . . . . . 8
1.3.4 Equipartition parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.5 Field and particle content of radio lobes and tails . . . . . . . . . . . . . 10
2 The X-ray environment of radio galaxies 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The thermal component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 The continuum spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Morphology of the X-ray emitting gas . . . . . . . . . . . . . . . . . . . 15
2.3 Radio source – environment interactions . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 X-ray cavities and their content . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Gas compression and heating by radio galaxies . . . . . . . . . . . . . . 18
i

3 Magnetic fields in the hot phase of the intergalactic medium 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Diffuse synchrotron sources in galaxy clusters . . . . . . . . . . . . . . . 22
3.1.2 Magnetic fields from diffuse radio sources . . . . . . . . . . . . . . . . . 24
3.1.3 Diffuse inverse Compton emission . . . . . . . . . . . . . . . . . . . . . 25
3.2 Faraday Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 Internal Faraday Rotation . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Foreground RM contributions . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.3 Galactic Faraday rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 RM analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 The magnetic field profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Tangled magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 The magnetic field power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7.1 Magnetic field power spectrum from two-dimensional analysis . . . . . . 38
4 Structure of the magneto-ionic medium around the Fanaroff-Riley Class I radio
galaxy 3C 449 43
4.1 The radio source 3C 449: general properties . . . . . . . . . . . . . . . . . . . . 43
4.2 Total intensity and polarization properties . . . . . . . . . . . . . . . . . . . . . 46
4.3 The Faraday rotation in 3C 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.1 Rotation measure images . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.2 The Galactic Faraday rotation . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5 Two dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5.2 Structure functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.6 Three-dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6.2 Magnetic field strength and radial profile . . . . . . . . . . . . . . . . . 60

4.6.3 The outer scale of the magnetic-field fluctuations . . . . . . . . . . . . . 64
4.7 Summary and comparison with other sources . . . . . . . . . . . . . . . . . . . 67
4.7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.7.2 Comparison with other sources . . . . . . . . . . . . . . . . . . . . . . . 70
5 Ordered magnetic fields around radio galaxies: evidence for interaction with the
environment 73
5.1 The Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.1 0206+35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.2 3C 270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.3 3C 353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.1.4 M 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Analysis of RM and depolarization images . . . . . . . . . . . . . . . . . . . . . 79
5.3 Rotation measure images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.5 Rotation measure structure functions . . . . . . . . . . . . . . . . . . . . . . . . 86
5.6 Rotation-measure bands from compression . . . . . . . . . . . . . . . . . . . . . 88
5.7 Rotation-measure bands from a draped magnetic field . . . . . . . . . . . . . . . 95
5.7.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.7.2 Axisymmetric draped magnetic fields . . . . . . . . . . . . . . . . . . . 96
5.7.3 A two-dimensional draped magnetic field . . . . . . . . . . . . . . . . . 97
5.7.4 RM reversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.8.1 Where do bands occur? . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.8.2 RM bands in other sources . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.8.3 Foreground isotropic field fluctuations . . . . . . . . . . . . . . . . . . . 100
5.8.4 Asymmetries in RM bands . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8.5 Enhanced depolarization and mixing layers . . . . . . . . . . . . . . . . 104
5.9 Conclusions and outstanding questions . . . . . . . . . . . . . . . . . . . . . . . 104

6 Faraday rotation in two extreme environments 109
6.1 The sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1.1 B2 0755+37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1.2 M 87 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Two-dimensional analysis: rotation measure and depolarization . . . . . . . . . . 111
6.2.1 0755+37: images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.2 M 87: images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Two-dimensional analysis: structure functions . . . . . . . . . . . . . . . . . . . 118
6.4 Preliminary conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Summary and conclusions 125
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5.1 The tailed source 3C 449 . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5.2 The lobed radio galaxies 0206+35, 3C 270, M 84 and 3C 353 . . . . . . . 126
6.5.3 The radio galaxies 0755+37 and M 87 . . . . . . . . . . . . . . . . . . . 127
6.6 General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.7 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Appendix 133
Data reduction and imaging of lobed radio galaxies 135
.1 Observations and VLA data reduction . . . . . . . . . . . . . . . . . . . . . . . 135
.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
.2.1 0206+35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
.2.2 0755+37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
.2.3 M 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Bibliography 143
Acknowledgments 153

Abstract
The existence of diffuse magnetic fields ofµG strength in the hot intergalactic medium is now well
established. Our knowledge about them has greatly improved over the last few decades, mainly
thanks to radio continuum observations, which have detected synchrotron emission from cluster
diffuse sources (halos and relics) and Faraday rotation of polarized emission from embedded
and/or background radio galaxies. Such fields are not thought to be dynamically significant, since
they provide typical magnetic pressures one or two orders of magnitude below thermal values.
However, they are believed to strongly influence the heat conductivity in the intergalactic medium
and to inhibit the spatial mixing of gas and propagation of cosmic rays. Therefore, in order to
improve our knowledge of the physical processes in the gaseous environment of galaxies, accurate
measurements of quantities such as magnetic field strength, spatial variation, topology and power
spectrum are crucial. While most work until recently has been devoted to rich clusters of galaxies,
little attention has been given to sparser environments, such as groups of galaxies, although similar
physical processes are likely to be at work.
The purpose of this thesis is to investigate the strength and structure of the magnetized medium
surrounding radio galaxies via observations of the Faraday effect. This study is based on an
analysis of the polarization properties of radio galaxies selected to have a range of morphologies
(elongated tails, or lobes with small axial ratios) and to be located in a variety of environments
(from rich cluster core to small group). The targets include famous objects like M 84 and M 87. A
key aspect of this work is the combination of accurate radio imaging with high-quality X-ray data
for the gas surrounding the sources.
Although the focus of this thesis is primarily observational, I developed analytical models and
performed two- and three-dimensional numerical simulations of magnetic fields.
The steps of the thesis are: (a) to analyze new and archival observations of Faraday rotation
measure (RM) across radio galaxies and (b) to interpret these and existing RM images using
sophisticated two and three-dimensional Monte Carlo simulations.
i

This thesis aims to pose and answer the following basic questions.
1. What is the origin of the bulk of the Faraday effect observed across radio galaxies? Two
possible contributors have been discussed in the literature: the intergalactic medium as a
whole and a local sheath formed through mixing layer surrounding the radio lobes.
2. How does the intergalactic magnetic field strength scale with the thermal density? Is there
a connection with the richness of the environment? Is the magnetic field ever energetically
important?
3. Can the intergalactic magnetic field always be described as an isotropic, Gaussian random
variable? If so, what is its power spectrum? Is there evidence for Kolmogorov turbulence?
What are the maximum and minimum scales? Is there evidence for a preferred direction or
two-dimensionality?
4. How do embedded radio galaxies affect the surrounding plasma and the structure and
strength of the field within it? What can be ascribed to the source morphology? Recent X-
ray observations have detected regions of low emissivity (“cavities”) in the X-ray emitting
gas at the position of the radio lobes. It appears that the radio sources have displaced the
surrounding thermal medium rather than mixing with it. What might be the consequences
for the magnetic field structure?
5. What is the content of the radio lobes and hence of cavities? In some cases, the relativistic
plasma in the radio lobes can provide enough pressure to support them, but others require
an additional pressure component – most likely from entrained and heated (kT> 10 keV)
intergalactic medium. This heated component should be detected through polarization
studies since it may cause internal Faraday rotation.
The approach has been to select a few bright, very extended and highly polarized radio galaxies.
This is essential to have high signal-to-noise in polarization over large enough areas to allow
computation of spatial statistics such as the structure function (and hence the power spectrum) of
rotation measure, which requires a large number of independent measurements. New and archival
Very Large Array (VLA 1 ) observations of the target sources have been analyzed in combination
with high-quality X-ray data from the Chandra, XMM-Newton and ROSAT satellites. The work
has been carried out by making use of:
1. Analytical predictions of the RM structure functions to quantify the RM statistics and to
constrain the power spectra of the RM and magnetic field.
1 The Very Large Array is a facility of the National Science Foundation, operated under cooperative agreement by
Associated Universities, Inc.

2. Two-dimensional Monte Carlo simulations to address the effect of an incomplete sampling
of RM distribution and so to determine errors for the power spectra.
3. Methods to combine measurements of RM and depolarization in order to constrain the
magnetic-field power spectrum on small scales.
4. Three-dimensional models of the group/cluster environments, including different magnetic
field power spectra and gas density distributions.
The thesis is organized as follows: Chapters 1–3 are introductory, while 4–6 contain the original
work developed in the thesis.
Chapter 1 briefly outlines the essential physics required to understand the polarized emission and
energetic of radio galaxies.
The gaseous environment of radio galaxies and implications for their interaction are described in
Chapter 2.
Chapter 3 reviews the known properties of intergalactic magnetic fields and outlines different
methods of analysis.
Chapter 4 presents a study of the magnetic field in the hot medium surrounding the radio galaxy
3C 449, located in the centre of a nearby galaxy group. High quality RM and depolarization images
have been produced by making use of archival VLA data at seven frequencies in the range 1365
– 8385 MHz. Structure-function and simulation techniques have been used to model the magnetic
field power spectrum over a wide range of spatial scales and to estimate both the minimum and
maximum scale of the magnetic field variations. The central field strength and its dependence on
density have been constrained. The work of this Chapter has been published in Guidetti et al.
(2010), reported in the section Highlighted papers of the journal Astronomy & Astrophysics.
Chapter 5 concerns the analysis of the RM of the nearby radio galaxies 0206+35, 3C 270, 3C 353,
and M 84. This represents the most innovative part of the thesis. The sources are embedded in
different environments, but show the same double-lobed radio morphology. They all show highly
anisotropic RM structures (RM bands) which are clearly different from the isotropic variations
seen in previously-published RM images. This is a new RM phenomenon and a first attempt to
interpret it as a consequence of source-environment interaction is presented. Analytical models for
the magnetic field and simulations of the RM expected from expanding sources in the magnetized
intergalactic medium are developed. This approach is entirely new and the main results are that a
simple compression mechanism cannot produce all the observed properties of the RM bands, and
a two-dimensional draped magnetic field provides a much better description of the data. The work
of this Chapter will be published in Guidetti et al. (2011, in press).

Chapter 6 presents preliminary results from two-dimensional analyses of the polarization of two
radio galaxies embedded in extremely different environments: 0755+37 in a very poor group and
M 87 at the centre of the cool core Virgo cluster. The work of this Chapter will be published in
two forecoming papers: Guidetti et al. (in preparation), Guidetti et al. (in preparation).
Chapter 6.4 summarizes all the results presented and briefly lists some topics for further work.
It is pointed out that this thesis has shown that the magnetized medium surrounding radio
galaxies appears more complicated than was apparent from earlier work. Three distinct types
of magnetic-field structure are identified: an isotropic component with large-scale fluctuations,
plausibly associated with the intergalactic medium not affected by the presence of a radio source;
a well-ordered field draped around the front ends of the radio lobes and a field with small-scale
fluctuations in rims of compressed gas surrounding the inner lobes, perhaps associated with a
mixing layer.
In the Appendix new VLA polarization data for the nearby radio galaxies 0206+35, 0755+37 and
M 84 are presented. This work will be published in the paper Laing, Guidetti et al. (2011, in prep).
Throughout the thesis I assume a cosmology with H 0 = 71 km s −1 Mpc −1 ,Ω m = 0.3, andΩ Λ = 0.7.

Chapter 1
Physics of radio galaxies
In this Chapter I will outline the essential physics required to understand the polarized
emission of radio galaxies and their interactions with the local environment. Sec. 1.1
introduces radio sources in the context of active galaxies in general, and Sec. 1.2 discusses
their principal emission mechanisms. The morphologies of radio galaxies, the physical parameters
of their emitting regions and the implications for their interaction with the environment are
described in Sec. 1.3.
1.1 Active galaxies
Active galactic nuclei (AGN) are the most powerful, persistent sources of luminosity in the
Universe. Their luminosities range from about 10 40 up to 10 47 ergs s −1 and their emission is spread
widely across the whole electromagnetic spectrum. AGN are so called because of the current
accepted model explaining their nature. They are believed to be powered by accretion onto supermassive
black holes (∼ 10 7 − 10 10 M ⊙ ) located at the nuclei of so-called “active galaxies”. Indeed,
the radiation emitted by such nuclei cannot be explained just by starlight or normal supernova
activity and can exceed the total emission of the rest of the galaxy by many order of magnitude.
Moreover, the AGN emission is often variable on time-scales ranging from years down to hours
or even minutes. Causality arguments imply that an object varying in a time t must be smaller
than the light-crossing time ct (where c is the speed of light in the vacuum) and therefore must be
spatially small. Accretion mechanisms onto a such small object with mass≃ 10 8 M ⊙ can efficiently
convert potential and kinetic energy to radiation and bulk outflow, and therefore account for high
luminosities and their rapid variations.
Since luminosity excesses have been observed across the entire electromagnetic waveband,
there is no single observational signature of active galaxies, which can be classified in many
1

2 1. Physics of radio galaxies
ways. This has often lead to a confusing terminology. The distinctions between different types of
AGN reflect historical differences in how objects were discovered or initially classified, rather
than real physical differences. Although the spectrum of these objects can extend across the
whole electromagnetic spectrum, the relative intensity between different wavebands and spectral
lines features are extremely different and provide a basic classification for the AGN (e.g. radio
galaxies, quasars, Seyfert galaxies, blazars). AGN are conventionally classified as either “radioloud”
or “radio-quiet”, depending on the ratio of their luminosities in the radio and optical bands.
Physically, this distinction reflects the relative importance of relativistic jets and their associated
non-thermal emission compared with radiation directly related to the accretion process. Jets are
an important, and often dominant, energy loss channel for radio-loud AGN. The cause of the
difference between radio-loud and radio-quiet AGN is a complicated and unresolved issue, which
may involve the spin of the central black hole (e.g. Meier 2002) or the details of the accretion
process.
A major simplification for radio-loud and radio-quiet objects is the “unified model”, which
suggests that different classes of AGN are in fact the same objects seen at different angles to the
line of sight. There are two physical mechanisms: beaming of radiation from relativistic jets
(radio-loud only) and obscuration of the central regions of the AGN by a dusty torus (e.g. Barthel
1989; Antonucci 1993; Urry & Padovani 1995).
Since this thesis is based on the analysis of the polarization properties of extended radio
galaxies, in the next sections I will concentrate on the main non-thermal emission mechanisms
observed from radio-loud active galaxies and their implication about the energy content of radio
galaxies.
1.2 Non-thermal emission mechanisms of active galaxies
The two main non-thermal radiation processes in radio-loud active galaxies are synchrotron
radiation and Inverse-Compton (IC) scattering. Their relative importance depends on the
observing frequency: IC emission from a given population of relativistic electrons is emitted at
higher frequencies than the corresponding synchrotron component. For example, in the extended
lobes of radio galaxies, synchrotron emission dominates at radio frequencies and IC in the X-ray
band.
1.2.1 Synchrotron radiation
Synchrotron emission is generated by the acceleration of relativistic charged particles in a
magnetic field B. When detected, it provides therefore the more direct way to detect magnetic
2

1.2. Non-thermal emission mechanisms of active galaxies 3
fields in astrophysical sources.
Lorentz factorγ(electron energyǫ=γm e c 2 ) is:
dǫ
dt
The synchrotron power emitted by a relativistic electron with
[ erg
]
≃ 1.6×10 −3 (B [µG] sinθ) 2 γ 2 (1.1)
s
where θ is the pitch angle between the electron velocity and the magnetic field direction.
This equation illustrates the degeneracy between particle energy and magnetic field: a given
synchrotron power can be produced by a highly energetic particle in a low magnetic field or vice
versa.
The spectral distribution for a single electron can be assumed to be approximately
monochromatic since it peaks sharply at a frequency:
ν c [GHz]≃4.2×10 −9 γ 2 (B [µG] sinθ). (1.2)
From Eq. 1.2, it is derived that electrons ofγ≃10 4 in magnetic fields of B≃10µG produce
synchrotron radiation in the radio band (ν c ∼ 4 GHz), whereas electrons ofγ ≃ 10 7−8 in the
same magnetic field radiate in the X-rays. A representative example is given by the nearby active
galaxy Pictor A, where the spectrum of the jet from radio to X-ray wavelengths can be described
as synchrotron emission from a population highly energetic particles with a distribution of Lorentz
factors (Wilson, Young & Shopbell 2001).
For a homogeneous population of electrons with an isotropic pitch-angle distribution and a
power-law energy spectrum with the particle density between energiesǫ andǫ+dǫ given by
N(ǫ)dǫ= N 0 ǫ −δ dǫ, (1.3)
the total intensity spectrum, in optically-thin regions has the functional form:
S (ν)∝ν −α , (1.4)
where the spectral indexα=(δ−1)/2. This spectrum of Eq. 1.4 is described as non-thermal,
since the energy spectrum of the emitting particles is not Maxwellian - i.e. it does not have a
single temperature.
1.2.2 Polarization of synchrotron radiation
In the optically-thin case, for a homogeneous and isotropic distribution of relativistic particles
whose energy distribution follows the power law of Eq. 1.3 and in a uniform magnetic field, the
3

4 1. Physics of radio galaxies
intrinsic degree of polarization has the value (Longair 1994):
P Int = δ+1
δ+ 7 , (1.5)
3
and the electric vector is perpendicular to the projection of the magnetic field onto the plane
of the sky.
For typical values of the index of the energy spectrumδ = 2.5, the intrinsic
polarization degree in an uniform field is expected to be≈70%, which represents the maximum
allowed value.
they exceed 0.3-0.4.
Degrees of polarization≥ 0.1 are often observed at 1.4 GHz; less commonly
A reduction of the expected degree of polarization might stem from an
intrinsically complex magnetic field structure within the source and/or depolarization effects, as
discussed in Secs. 3.2.1 and 3.3. If the magnetic field can be expressed as the superposition of
two components, one uniform B u , the other isotropic and random B r , the observed degree of
polarization is approximately given by (Burn 1966):
B 2 u
P Obs = P Int
B 2 u+B 2 . (1.6)
r
which gives the ratio of the energy in the uniform field over the energy in the total field, and is an
indicator of the field uniformity and structure. However, random magnetic fields are likely to be
anisotropic, in which case Eq. 1.6 does not hold.
A mathematically convenient way to describe the linear polarization state is given by the
Stokes parameters I, Q and U (Stokes, 1852).
The polarized intensity and the polarization angle of linearly polarized radiation can be
described by:
P λ = (Q 2 λ + U2 λ )1/2 (1.7)
and
Ψ λ = 1 2 arctan (
Uλ
Q λ
)
, (1.8)
Images of polarized intensity P, degree of polarization P/I and position angleΨ(and in turn
magnetic field) can be derived across radio galaxies from the I, Q, and U images through radio
observations in full polarization mode.
Once corrected for the Faraday rotation (Sec. 3.2) the projected magnetic field in the plane
of the sky appears to be tangential to the edges of the lobes in both low- and high-power radio
sources. The field orientation in the jets differs between the two classes. Low-power jets usually
show magnetic field parallel to the jet axis close to the core with a transition to perpendicular field
as the jet expands. In contrast, in powerful sources the magnetic field is usually parallel to the jet
axis for the whole length of the jets.
4

1.2. Non-thermal emission mechanisms of active galaxies 5
Therefore, the measurement of synchrotron emission from radio galaxies provides
information about the index of the energy distribution of particles and the strength of magnetic
fields inside the source, while the degree of polarization is an important indicator of the field
uniformity and structure. The high degrees of linear polarization observed from radio-galaxy jets
and lobes make them ideal probes of the foreground magnetised medium (Sec. 3.2).
1.2.3 Inverse Compton emission
Relativistic electrons in a radiation field can scatter low-energy photons to high energy through
the inverse-Compton (IC) effect. The reason for the adjective “inverse” is that the electrons lose
energy rather than the photons as in the usual Compton scattering. IC scattering increases the
frequency of the scattered photonsν ph by a factor 4 3 γ2 , whereγis the Lorentz factor of the
relativistic electrons (e.g. Rybicki & Lightman 1986). The low-energy scattered photons are
often dominated by the ubiquitous 3K cosmic microwave background (CMB). In the presence
of relativistic particles withγ∼10 3−4 , CMB photons are scattered from the original frequency
around 10 11 Hz to about 10 17−18 Hz, corresponding to the X-ray andγ-ray domains (0.8- 20 keV).
Since the power radiated via the IC process by an electron has the same functional dependence
on the electron energy as in Eq. 1.1, if the synchrotron and IC emission originate from the same
relativistic electron population their fluxes are related. For the electron energy distribution of Eq.
1.3, the two spectra share the same spectral indexα. The spectral index relates to the photon index
of the IC emission asΓ X =α+1.
Given that the synchrotron emissivity is proportional to the magnetic energy density U B , while
the IC emissivity is proportional to the energy density in the photon field U ph , it follows that:
S syn
S IC
∝ U B
U ph
, (1.9)
where S syn and S IC are the synchrotron and IC fluxes, respectively.
From the ratio between the IC and synchrotron fluxes, in principle one can derive an
estimate of the total magnetic field, averaged over the emitting volume. In terms of observational
parameters this is:
B[µG] 1+α S syn (ν r )[Jy]
= h(α)
S IC(E1 −E 2 )[ergs −1 cm −2 ] (1+z)3+α (0.0545ν r [MHz]) α × (1.10)
× (E 2 [keV] 1−α − E 1 [keV] 1−α ,
where S syn(νr ) is the synchrotron flux at the radio frequencyν r and the flux S IC(E1 −E 2 ) is
integrated over the energy interval E 1 − E 2 .
5

6 1. Physics of radio galaxies
1.3 Radio galaxies
Radio galaxies are radio-loud AGN hosted by massive early type galaxies (visual magnitude
M v < -20), with radio powers at 408 MHz spanning the range 10 23−28 WHz −1 . Their radio spectra
have approximately power-law forms with typical spectral indicesα=0.8±0.2, consistent with
synchrotron emission from relativistic particles with power-law energy spectra (Sec. 1.2.1). The
electron Lorentz factors areγ≥100 and magnetic field strengths are∼µG. The synchrotron origin
of the radio emission is confirmed by the smooth broad-band nature of the emission (10 MHz-
10 GHz) and the high degree of linear polarization (commonly≥0.1 averaged across the source,
but reaching values of 0.7 in sub-regions).
The energy source for the relativistic particles and magnetic fields is thought to be the central
black hole at the nucleus of the radio galaxy. The energy is transferred from this engine outwards
by more or less collimated, initially relativistic, flows or beams, whose visible manifestations are
the radio jets.
1.3.1 Radio-galaxy morphologies
Radio galaxies show a wide range of structures and linear sizes, going from a few ten of pc
up to Mpc (Bridle & Perley 1984; Laing 1993), and hence they can be more extended than the
parent galaxy. The main morphological classification of extended radio galaxies is that made by
(Fanaroff & Riley 1974). This was based on the sharp change in morphology of the sources in
the 3C catalogue around the radio power 10 24.5 WHz −1 at 1.4 GHz, close to the break in the radio
luminosity function. Fanaroff & Riley pointed out that low-power sources tend to be brightest close
to their nuclei (“edge-darkened”) whereas high-power ones are brightest at their outer extremities
(“edge-brightened”). These are respectively classified as Fanaroff-Riley I and Fanaroff-Riley II
sources, (or FR I and FR II sources). The prototypes of FR I and FR II sources are respectively
3C 31 and Cygnus A (Fig. 1.1). The sources discussed in detail in this thesis are mostly members
of the FR I class.
The main morphological components of FR I and FR II radio galaxies observed with
arcsecond resolution (e.g Bridle & Perley 1984) are as follows.
• The “core” is an unresolved component coincident to within the observational errors with
the optical nucleus. It represents the partially optically thick base of the jets (see below).
Cores are relatively stronger in FR I sources.
• The “jets” are long narrow features, which emerge from the core and propagate generally in
opposite directions. FR I jets are typically bright, with wide opening angles, while those of
6

1.3. Radio galaxies 7
FR II sources appear weak and well collimated, suggesting a more efficient energy transport
than in FR Is. FR I jets are thought to decelerate to sub-relativistic speeds on kpc scales,
while FR II jets remain relativistic over their whole lengths.
• The “hot-spots” are bright and compact regions observed only in FR II sources, close
to the extremities of the radio structures. They are interpreted as the termination or
major disruption of the jets at strong shocks (which requires that FR II jets are internally
supersonic).
• The “lobes” are wide structures with small axial ratios which lie on either side of the
parent galaxy and are often aligned across the nucleus on scales up to Mpc. Most of the
radio emission is therefore seen between the end of the jets and the core. Synchrotron
spectra of lobes show steepening towards the nucleus and it is therefore likely that lobes are
formed from relativistic particles left behind at or back-flowing from the region where the
jet impacts the external medium. Lobes are found in both FR I and FR II sources.
• The “tails” are elongated and sometimes irregular structures, mostly farther from the nucleus
than the end of a jet. Their synchrotron spectra steepen away from the nucleus, suggesting
that they are flowing outwards. They are found almost exclusively in FR I sources.
1.3.2 Morphology and expansion of lobes and tails
The physics of the interaction between a radio source and the surrounding IGM appears to be
significantly different for lobes and tails. Also, as will become apparent in later Chapters, the
magnetization of the IGM around a radio source appears to depend on the morphology of the
extended emission. For these reasons, I briefly discuss the differences between lobes and tails.
The richness of the environment may play a role in the formation of tails rather than lobes.
Most of the tailed sources are observed in galaxy clusters or rich groups and often show distorted
morphologies which are likely to be caused by gas sloshing in the potential well of the cluster or
by the high ram pressure exerted by the thermal gas on fast-moving galaxies (respectively Wideangle
tails, e.g. 3C 465, Eilek et al. 1984 and Narrow-angle tails, e.g. NGC 1265, O’Dea & Owen
1986). There is therefore clear evidence for jet/environment interaction in tailed sources. Indeed,
tails are likely to be formed if the jet entrains enough material to slow it to below the external
sound speed and/or if buoyancy causes the synchrotron plasma to be pushed outwards. For both
these reasons, FR I tails are thought to be expanding sub-sonically, basically buoyantly, and in
pressure equilibrium with their surroundings.
Lobes or bridges, on the other hand, do sometimes show evidence for shocks in the
surrounding X-ray emitting gas, suggesting that they are expanding supersonically (Sec. 2.3.2).
7

8 1. Physics of radio galaxies
So far, the best examples of strong shocks surrounding bridges are found only in two FR I sources
(Centaurus A, Kraft et al. 2003; NGC 3801, Croston et al. 2007). Instead, weak shocks appear to
occur more frequently and they are seen in both FR I (e.g. Perseus, Fabian et al. 2003; Hydra A,
Nulsen et al. 2005a; M87, Forman et al. 2005) and FR II sources (e.g. Cygnus A, Wilson, Smith &
Young 2006; Hercules A, Nulsen et al. 2005b; 3C 444, Croston et al. 2010). Therefore, standard
models of supersonic radio-lobe expansion, which were originally thought to apply only to FR II
sources (e.g. Scheuer 1974; Kaiser & Alexander 1997), are also likely to describe lobes in FR I’s.
It may be that the expansion of FR I lobes is supersonic only in the forward direction (driven
by the ram pressure of the jets) and that the lobes are in static pressure equilibrium with their
surroundings closer to the galaxy.
To conclude, the factors which determine the large-scale morphology of radio galaxies have
not been disentangled yet, although the initial speeds of the jets and their density contrast with the
external medium both seem to be important. Whether a lobe or a tail is formed clearly makes a
substantial difference to the interaction between a source and its surroundings.
1.3.3 Internal physical conditions of radio galaxies
In order to understand the interaction of radio galaxies with their environments, we need to
quantify the internal conditions in the extended emitting regions (lobes and tails). From the
synchrotron emissivity alone, it is not possible to derive unambiguously the energy of the
relativistic particles and magnetic field, because of the degeneracy between them (Eq. 1.1).
Constraints on the energy density in relativistic electrons and magnetic field can be derived from
synchrotron emission if IC emission from the same electron population is observed, in which
case the degeneracy between particles and fields can be resolved (Eq. 1.10). The sum of particle
and field energy densities can then be compared with the thermal pressure in the surroundings.
If IC emission is lacking, in order to separate the magnetic and relativistic particles densities,
one must make some assumptions about the relation between electrons and magnetic field. One
commonly used approach is to assume energy equipartition between them: this is described in the
next section.
1.3.4 Equipartition parameters
In the context of radio sources, the term “energy equipartition” is commonly used to mean that the
energy densities in relativistic particles and magnetic field are equal. This is an assumption, with
no rigorous justification, but has some empirical support from observations of FR II lobes. It is
also close to the condition that the total energy density in particles and field is a minimum.
8

1.3. Radio galaxies 9
(a)
(b)
Figure 1.1: Examples of extended radio galaxies with lobes or plumes. Top panel: bridged radio
galaxy Cygnus A (FR II) in a cluster of galaxy at z=0.056. This is the brightest extragalactic
radio source in the sky. Bottom panels: montage by A.H. Bridle showing the tailed radio source
3C 31 (FR I, in red) at the center of a galaxy group at z=0.0169, superimposed on Hubble Space
Telescope WFPC2 image (a) and on DSS image (b).
The field strength and pressure corresponding to the equipartition condition (e.g. Longair
1994) are:
B eq [µ G] ≃ 0.9·L 2/7
[ergs −1 ] V−2/7 [kpc 3 ] ζ2/7 Φ −2/7 (1.11)
P eq [dyne/cm 2 ]∝L 4/7
[ergs −1 ] V−4/7 [kpc 3 ] ζ4/7 Φ −4/7 , (1.12)
where L and V are the radio luminosity and the volume of the source,ζ is the ratio of the total
energy in particles to the energy in electrons alone, andΦis the so called “filling factor”, which
is the fraction of the total source volume occupied by emitting material. For a relativistic plasma
Γ=4/3, so the total minimum energy density of the source is u min ≈ 3× P eq . Since P eq represents
the pressure exerted by the synchrotron-emitting plasma, throughout the thesis it will be reported
as P syn .
The comparison of the pressure P eq with the external one provided by the surrounding
9

10 1. Physics of radio galaxies
medium (Chapter 2) is useful to understand the equilibrium conditions of radio galaxies. In both
FR I and FR II radio sources, B eq and P syn are found to be a fewµG and 10 −12 −10 −13 dyne/cm 2 ,
respectively.
However, all the equipartition estimates must be taken with caution, because each of the
variables in Eqs. 1.11 and 1.12 is potentially a source of uncertainty. Indeed, the filling factor
Φ is unknown and often assumed to be 1 although in some radio galaxies there is evidence for
filamentary structure. Some assumptions on the geometry of the source must also be made in
order to compute the source volume V. The ratioζ is also subject to major uncertainties, since
the energy density in relativistic protons is unconstrained. Moreover, the equipartition parameters
are strongly related to the functional form of the particle energy spectrum and therefore to the
synchrotron emission spectrum. Good constraints on the energy spectrum require wide frequency
coverage. Indeed, even small changes in the lower limit of the relativistic particle energy can
have a huge impact on the determination of B eq and P min . This is particularly true for steep radio
spectra, where low energy particles contribute most of the total energy.
My personal conclusion is that minimum energy arguments are of limited usefulness. The best
use of these estimates lies in making the most conservative assumptions and treating the resulting
pressures as lower limits.
1.3.5 Field and particle content of radio lobes and tails
FR I radio galaxies with tails are believed to expand sub-sonically in pressure equilibrium with
the surroundings, and to rise buoyantly in the intergalactic medium. On the other hand, some
FR I sources with bridges appear to be surrounded by shocked X-ray emitting gas, which suggests
supersonic expansion.
As already pointed out, when IC emission is detected from the radio sources, then the total
energy density in relativistic electrons and fields can be estimated and compared with the external
pressure, also derived from X-ray observations. For some FR II sources, this has been done
(e.g. Kataoka & Stawarz 2005; Croston et al. 2004; Laskar et al. 2009; Isobe et al. 2011) The
conclusion is that most FR II lobes are close to the minimum energy condition, in the sense that
relativistic electron and field energy densities are comparable, and that their sum is in turn close
to the external pressure, as expected for static thermal confinement (e.g. Hardcastle et al. 2002;
Croston et al. 2004; Konar 2009). So far, there are no unambiguously detections of IC emission
from FR I lobes. Instead, several studies have shown that the synchrotron equipartition pressure
P syn within the FR I radio lobes is often an order of magnitude smaller than the external pressure
exerted by the hot surrounding gas (e.g. Morganti et al. 1988; Worrall & Birkinshaw 2000; Blanton
et al. 2001; de Young 2006; Croston et al. 2003, 2008). Without a further source of internal
10

1.3. Radio galaxies 11
pressure, the radio sources would then collapse.
To solve the pressure balance problem for FR I sources we need to consider:
1. deviations from equipartition or
2. an additional source of pressure which is not detectable by current radio or X-ray
observations.
Deviations from equipartition in the sense of electron dominance could be given by a large excess
of low-energy electrons. These would be detectable via their IC radiation in at least some cases
(e.g. Croston, Hardcastle & Birkinshaw 2005). Therefore deviations from equipartition would
have to be in the direction of magnetic dominance.
An additional source of pressure could be provided by relativistic protons. These would have
to be accelerated in the lobes, since a relativistic proton population in the jets would exert too
high a pressure to allow collimation. Even cold protons cannot be transported by the jets in large
enough numbers without violating constraints on the mass flux (Laing & Bridle 2002a). The best
candidate to solve the pressure balance is therefore heated and entrained thermal material. This
would have to be hot (kT≥10 keV, e.g. Nulsen et al. 2002) but tenuous, to provide enough pressure
without radiating sufficiently to erase the soft X-ray depressions (“cavities”), often observed at the
position of radio lobes (Sec. 2.3.1).
11

12 1. Physics of radio galaxies
12

Chapter 2
The X-ray environment of radio
galaxies
2.1 Introduction
E
arly Einstein X-ray observations discovered that early-type galaxies, groups and
clusters of galaxies are spatially-extended X-ray sources with luminosities in the range
10 39−43 erg s −1 (e.g. Forman et al. 1979; Kriss et al. 1980; Biermann, Kronberg &
Madore 1982). Subsequent more sensitive and detailed X-ray images provided by the ROSAT,
Asca, XMM-Netwton and Chandra satellites have led to the detection of components on several
scales from several distinct emission mechanisms. These are:
• the end products of stellar evolution (point-like sources);
• emission spatially coincident with the bases of radio jets;
• the hot phase of the interstellar medium (ISM) of the galaxies and
• the hot and diffuse intergalactic medium (IGM) in groups and clusters of galaxies.
It is now well-established that the diffuse X-ray emission arises from hot and dilute plasma
emitting via thermal bremsstrahlung (Sec. 2.2.2).
Radio emission from active galaxies therefore coexists with a complex, hot medium, with
which it interacts in several ways.
• Radio jets can entrain the external medium, which is thought to be able to decelerate the
initially relativistic flows in FR I sources on typical scales of a few kpc (e.g. Laing & Bridle
2002a).
13

14 2. The X-ray environment of radio galaxies
• Both the lateral expansion of a radio source and the propagation of radio jets can do work
against the external hot gas, displacing and compressing it. Energy can also be transferred,
for example by shock heating or dissipation of sound waves.
• This transferred energy can prevent the cooling of hot gas and therefore suppress cooling
flows and star formation in massive galaxies (e.g. Sarazin 1988; Binney & Tabor 1995;
Mathews & Brighenti 2003).
• Conversely, hot gaseous atmospheres confine the radio emission, influencing its
morphology.
Therefore, X-ray observations of the hot gas are crucial to an understanding of the dynamics
and energetics of radio galaxies as well their impact on their environment. X-ray imaging can
show signatures of interactions between the radio galaxies and hot gas, while X-ray spectroscopy
can shed light on the mechanisms by which energy is transferred from the radio galaxy to the
surrounding plasma.
2.2 The thermal component
2.2.1 The continuum spectrum
Any hot, fully-ionized plasma with a temperature above 10 4 K emits via thermal bremsstrahlung,
which results from the acceleration of free electrons deflected in the Coulomb field of a ion.
The thermal bremsstrahlung emissivity at frequencyνfor an optically-thin plasma in collisional
equilibrium 1 can be expressed as :
J X (ν)∝n e n ions g(ν, T)T 1/2 exp −hν/K BT ergs −1 cm −3 Hz −1 sr −1 (2.1)
where n e and n ions are the electron and ion number densities and T is the gas temperature. 2
g(ν, T) ∝ ln(k B T/hν ≈ 1) is the Gaunt factor, which corrects for quantum mechanical effects
and for the effects of distant collisions. The square-law density behavior in Eq. 2.1 reflects the
collisional nature of the process. The emission of X-rays from ISM and IGM is well-described by
Eq. 2.1.
X-ray data can therefore be used to derive the physical properties of the X-ray emitting gas
such as central density, temperature profile and total mass. Fits to the X-ray spectra give average
1 Collisional equilibrium takes place when processes of electron ionization are exactly balanced by recombination
processes.
2 The electrons and ions are thought to have a Maxwellian distribution with a common temperature T .
14

2.2. The thermal component 15
Figure 2.1: Panel (a): Chandra observation of the massive (regular) cluster of galaxies A 1689
(blue) superimposed on the optical image (yellow, Hubble Space Telescope). Panel (b): fitting of
the surface brightness profile with a doubleβ model described in the inset. Taken from Xue & Wu
(2002).
gas temperatures increasing with the size and the mass of the environment. Typical temperatures
for clusters, groups and individual galaxies are given in Table 2.1.
2.2.2 Morphology of the X-ray emitting gas
Imaging of the X-ray emission from massive ellipticals, groups and clusters of galaxies has shown
that the morphologies of the emitting gas are quite heterogeneous, but that they can be related to
the dynamical state of the systems. Much of work has been carried out for galaxy clusters, but the
conclusions can be extended to sparser environments such as groups of galaxies and field galaxies,
which can be regarded as scaled-down version of rich clusters.
Forman & Jones (1982) first proposed a classification into “regular” and “irregular” X-ray cluster
morphologies, with a connection to the evolutionary state.
Essentially, regular clusters are those which show approximately round, centrally condensed
X-ray brightness distributions, decreasing smoothly outwards. Their temperatures and X-ray
luminosities are usually high and they often host a central dominant galaxy. Galaxy clusters
belonging to this class are thought to be evolved systems which have undergone dynamical
relaxation.
When the X-ray emission can be approximated as regular, its surface brightness profile is well
parametrized by the function:
] −3β+1/2
S (r)=S (0)
[1+ r2
(2.2)
15
r 2 c

16 2. The X-ray environment of radio galaxies
where r is the projected radial distance from the centroid of the X-ray surface brightness
distribution, S (0) is the central surface brightness, r c is the X-ray core radius andβ= µm pσ 2 r
k B T
. Here,
µ is the mean molecular weight, m p is the proton mass, andσ r is the radial velocity dispersion of
the galaxies.
The model of Eq. 2.2 has the advantage that its shape is uniquely described by only two
parameters: the core radius and the slopeβ. The latter determines the sharpness of the turnover
beyond the core radius and the asymptotic slope of the scaling. Physically,β represents the ratio
between the specific kinetic energy of the gravitationally bound galaxies and the specific thermal
energy of the hot gas. Fits to the X-ray surface brightness profiles generally giveβ∼0.4-0.7,
indicating that the energy per unit mass is higher in the hot gas than in the galaxies. The core
radius r c is highly correlated with the parameterβ, in the sense that larger r c corresponds to higher
β: indeed, a given X-ray surface brightness can be reproduced by a concentrated distribution with
a rapid decline (small r c and highβ), or by a more diffuse one with a flatter decline (large r c and
lowβ). Typical ranges for the fitted X-ray parameters are summarized in Table 2.1.
Assuming that the gas-density distribution is isothermal and hydrostatic, the surface
brightness profile of Eq. 2.2 corresponds to the density profile called aβmodel and given by
(Cavaliere & Fusco-Femiano 1976):
where n e (0) is the central electron density.
] − 3
n e (r)=n e (0)
[1+ r2 2 β
. (2.3)
Theβmodel adequately describes the surface brightness profiles of regular X-ray emission
over a wide range of radii. However, it fails in reproducing the strongly peaked X-ray surface
brightness distributions observed in some systems. This peak might be interpreted as emission due
to the ISM of the central elliptical galaxy, superimposed on that from the intergalactic medium, or
as an indication of the presence of a cooling flow (e.g. Trinchieri, Fabbiano & Kim 1997; Helsdon
& Ponman 2000; Croston et al. 2008). In both cases, the fit to the X-ray surface brightness is
significantly improved by using two distinctβmodels, one for each component (e.g. Croston et al.
2008). Analogously, fits to the temperature profile often require two-temperature models. Fig. 2.1
shows an example of X-ray emission from a relaxed galaxy cluster and the best fitting doubleβ
model describing the surface brightness profile.
Models of this type are used later in the thesis (Chapters 4 and 5) to parametrize the density
distributions of hot gas around radio galaxies.
r 2 c
16

2.3. Radio source – environment interactions 17
Table 2.1: Typical parameters describing the X-ray 1 emitting atmospheres in clusters, groups and
massive ellipticals.
environment L X n e (0) r c β kT
[erg s −1 ] [cm −3 ] [kpc] [keV]
cluster 10 43−45 10 −3 100-300 0.4-0.8 2-6
group 10 41−44 10 −3 − 10 −4 20-200 0.3-0.9 0.3-2
elliptical 10 39−41 0.1 5-20 ” 0.5-1.5
1 in the soft X-ray energy band≃0.2-10 keV.
Figure 2.2: Composite image of the galaxy cluster MS0735.6 7421 Hubble Space Telescope
(optical) superimposed on Chandra X-ray (blue) and Very Large Array at 330 MHz (red). The
X-ray data show the emission of hot gas at∼ 5× 10 7 K and two cavities, each roughly 200 kpc in
diameter coincident with radio lobes. Taken from McNamara et al. (2005).
2.3 Radio source – environment interactions
2.3.1 X-ray cavities and their content
An increasing number of clusters, groups and giant ellipticals show regions of X-ray surface
brightness depressions which appear as holes embedded in a brighter emission (see McNamara
& Nulsen 2007 for a review).
The surface brightness of such holes is around 20-40% below that of the surroundings. They
are commonly observed coincident with radio lobes and generally referred to as “cavities”. An
example is given in Fig. 2.2: the Chandra image displays the soft X-ray emission of the cluster
17

18 2. The X-ray environment of radio galaxies
MS0735.6 7421 with two large cavities, each roughly 20 kpc in diameter (McNamara et al. 2005).
Other spectacular examples of cavities associated with cluster sources are Hydra A (McNamara
et al. 2000), Centaurus A (Fabian et al. 2000) and Cygnus A (Carilli et al. 1994). Cavities are
observed also in group environments of FR I sources (e.g. Croston et al. 2008; Giacintucci et al.
2011; Gitti et al. 2010). The cavities observed so far have an average radius of 10 kpc (e.g. Bîrzan
et al. 2004), but there are examples with radii up to 100 kpc (Hydra A, Nulsen et al. 2005a).
The absence of soft X-ray emission from cavities shows that they cannot contain much thermal
plasma, unless it is much hotter than the surroundings (kT ≥ 10 keV). One possibility is that
cavities are filled entirely with synchrotron-emitting plasma associated with radio lobes (Nulsen
et al. 2002; Bîrzan et al. 2008 and reference therein), but the inference of an apparent pressure
deficit in the lobes of FR I radio galaxies (Sec. 1.3.5) suggests that there may also be a significant
amount of heated and entrained thermal plasma.
The energy required to inflate a cavity E cav adiabatically can be estimated roughly using only
X-ray observations. For cavities containing only relativistic mono-atomic gas, E cav = 4PV, where
P is the pressure internal to the lobe and V is the volume of gas displaced by the radio lobe. A
power-law relation is found between E cav and the radio-source luminosity. However, this relation
shows a large scatter (Bîrzan et al. 2004, 2008; Cavagnolo et al. 2010). E cav is a lower limit to
the energy supplied by the radio source, some of which may be dissipated (e.g. in shocks) or used
to inflate further undetected cavities (e.g. McNamara & Nulsen 2007). E cav estimates based only
on enthalpy range from 10 55 up to 10 61 erg s −1 , reaching the highest values in rich clusters. The
energy deposition rates are high enough to prevent gas cooling below 2 keV in several systems
(e.g. Bîrzan et al. 2008).
2.3.2 Gas compression and heating by radio galaxies
If radio lobes are expanding supersonically 3 , at least in the forward direction of the jets, then a
bow shock is expected to form ahead of them (Fig. 2.3). The shock velocity is essentially the
expansion velocity of the lobe. Shock waves are characterized by a discontinuous and very sharp
change in the characteristics of the gas. Density and temperature jumps across the shock surface
(Rankine-Hugoniot conditions) can be derived from conservation of mass, momentum and energy
across the shock. In a perfect gas and for an adiabatic shock traveling normal to the gas flow the
jump conditions for the densityρand temperature T (Landau & Lifshitz 1987) are:
ρ 2
ρ 1
=
γ+1
γ−1+ 2 M 2 (2.4)
3 with respect to the sound speed of the X-ray emitting diffuse gas.
18

2.3. Radio source – environment interactions 19
Bow Shock
Rlobe
1
2
3
4
Rshell
Radio Lobe
ISM
Shell
Figure 2.3: Schematic diagram of the regions around a supersonically expanding lobe in the
ISM. Region 1 is the radio lobe, region 2 the observed X-ray enhancement region, region 3 is
a physically thin layer where the Rankine-Hugoniot shock conditions are met, and region 4 is the
ambient ISM. The figure is taken from Kraft et al. (2003).
ρ 1
T 2
= 2γM2 −γ+1
(2.5)
T 1 γ+1 ρ 2
where the subscripts 1 and 2 stand for the unshocked and shocked gas, andMis the Mach
number of the shock wave.M is defined as the ratio between the shock velocity v and the sound
speed in the un-shocked gas:M=v/c 1 .
In the limit of a very strong shock (M≫1), the two ratios become:
ρ 2
= γ+1
ρ 1 γ−1
(2.6)
T 2
M 2
= 2γ(γ− 1)
T 1 (2γ+ 1) 2.
(2.7)
These show that for a very supersonic lobe expansion, T can be arbitrarily large, whereas the
density jump attains a finite maximum value, which is 4 for a thermal mono-atomic plasma (a
reasonable approximation for the hot IGM). The gas compression and heating take place over the
mean free path of the gas and hence the shock front is expected to be narrow.
The presence of shocks surrounding radio lobes can be confirmed using deep X-ray
observations by extracting surface brightness and spectra profiles. These allow the detection of
density and temperature jumps, respectively, (Eqs.2.3 and 2.2) and from these the shock Mach
numberMcan be estimated. So far the two known cases of strong shocks associated with
expanding radio-galaxy lobes are found in the FR I sources Cen A and NGC 3801. In these sources
the shocked hot gas is 10 and 4-5 times hotter than the surroundings, corresponding respectively
19

20 2. The X-ray environment of radio galaxies
-17 00 00
2 4 6 8
-17 00 00
30
30
DECLINATION (J2000)
01 00
30
02 00
DECLINATION (J2000)
01 00
30
02 00
3
2
1
Outer
30
30
03 00
22 14 32 30 28 26 24 22 20
RIGHT ASCENSION (J2000)
03 00
22 14 32 30 28 26 24 22 20 18
RIGHT ASCENSION (J2000)
Figure 2.4: (Left): Overlay of a 5 GHz VLA map (red) of 3C 444 on 0.5 - 5.0 keV Chandra
data (blue) indicating the relationship between the radio and X-ray structures, including cavities
coincident with the radio lobes, and a sharp elliptical surface brightness drop surrounding the
source. (right): temperature map in keV with 5 GHz contours overlaid. The spectral extraction
regions are numbered. The figure is taken from (Croston et al. 2010).
to shock Mach numbersMof about 8 and 4 respectively (Kraft et al. 2003; Croston et al. 2007).
In both the cases, the total energies stored in the cavities are several times larger than the estimated
PV work.
Weak shocks withM≈1.2−1.7 surrounding cavities associated with FR I and FR II sources are
observed more frequently (e.g. McNamara et al. 2005; Nulsen et al. 2005a; Croston et al. 2008,
2010). Fig. 2.4 shows the case of the FR II radio galaxy 3C 444 where the surrounding intra-cluster
medium is characterized by cavities and a temperature jump of a factor∼1.7, probably caused by
a spheroidal shock (Croston et al. 2010).
The effects of these radio-galaxy shocks on the magnetic field in the IGM are analysed in
detail in Chapter 5.
20

Chapter 3
Magnetic fields in the hot phase of the
intergalactic medium
In this Chapter I will briefly review the state of our knowledge of intergalactic magnetic
fields. Most attention is given to the results of the analysis of Faraday effect across radio
galaxies, on which this thesis is based, while those from other techniques are summarized
more briefly. In particular I will show that detailed radio observations of polarized radio galaxies
indicate the presence of turbulent magnetic fields fluctuating over a wide range of spatial scales
and provide a means of measuring their power spectra.
3.1 Introduction
The existence of magnetic fields in the extragalactic universe is now well established. Our
knowledge about them has greatly improved over the last few decades, mainly thanks to radio
continuum observations, which have detected magnetic fields atµG levels in objects such as galaxy
disks and halos and intergalactic media in both groups and clusters of galaxies. It is also possible
that intergalactic voids are permeated by a widespread magnetic field.
Despite their ubiquity, the role of extragalactic magnetic fields has often remained an ignored
aspect of astrophysics, because of their low energy densities. µG-strength magnetic fields are
now thought to be important for multiple reasons. Fields associated with the thermal IGM
are not thought to be dynamically significant, since they provide typical magnetic pressures
one or two orders of magnitude below thermal values. However, they are believed to strongly
influence the IGM heat conductivity, inhibiting the spatial mixing of gas and propagation of
cosmic rays(e.g. Balbus 2000; Bogdanović et al. 2009). Indeed, the IGM is so extremely dilute
that is characterized by huge collisional mean free paths (∼20 kpc). In this condition, even for
21

22 3. Intergalactic magnetic fields
a weak magnetic field, the expected Larmor radius (∼ 10 8 cm for T=10 8 K and B=1µG) is much
smaller than the mean free path. It follows that the charged particles are channeled only around
the magnetic field lines, so thermal conduction becomes anisotropic.
A detailed knowledge of the strength and structure (coherence lengths, fluctuations scales)
of intergalactic magnetic fields is therefore important to a better understanding of the physical
processes in the gaseous environment of galaxies.
Magnetic-field analysis has until recently been restricted to rich clusters of galaxies, while
little attention has been given in the literature to sparser environments, such as groups of galaxies,
although similar physical processes are likely to be at work. Intra-cluster magnetic fields have
been measured using different techniques (e.g. Carilli & Taylor 2002) based mainly on the study
of:
• diffuse radio synchrotron sources within clusters;
• inverse Compton emission and
• Faraday rotation of polarised radio sources both within and behind clusters.
These analyses have led to slightly discrepant estimates for the field strength, which is some
cases could be reconciled taking into account the several and different assumptions on which are
the methods are based.
3.1.1 Diffuse synchrotron sources in galaxy clusters
The most direct proof of presence of magnetic fields mixed with the ICM is provided by the
detection of diffuse radio synchrotron emission on scales up to Mpc in an increasing number of
galaxy clusters (see e.g. Ferrari et al. 2008 for a review). Since this emission appears not to be
associated with single active galaxies but to arise from the ICM itself, it indicates the presence
of both relativistic particles and large-scale magnetic fields in such plasma. These are globally
called the “non-thermal” component of galaxy clusters. Their radio spectra are well explained as
synchrotron emission from ultra-relativistic electrons (Lorentz factor> ∼ 1000) moving in magnetic
fields of 0.1-1µG strength.
Diffuse cluster radio emission shows two types of morphology: radio relics and halos. While
relics are irregular and polarized structures observed at the periphery of clusters, with size ranging
from few tens of kpc up to 10 3 kpc (e.g. Harris et al. 1993; Clarke & Enßlin 2006; Bagchi et
al. 2006; Bonafede et al. 2009a; van Weeren et al. 2010), radio halos are regular, distributed
as the thermal X-ray emission of the ICM and typically showing low fractional polarization
22

3.1. Introduction 23
Figure 3.1: Example of a relic. Left: Westerbork Synthesis Radio Telescope (WSRT) image at
1.4 GHz overlaid on the X-ray emission from ROSAT showing the hot ICM (red contours). Right:
Polarization electric field vectors at 4.9 GHz (corrected for the effects of Faraday rotation) and
with lengths proportional to the fractional polarization at the same frequency. A reference vector
for 100% polarization is drawn in the top left corner. Both figures are taken from van Weeren et
al. (2010).
(e.g. Giovannini & Feretti 2002; Clarke & Enßlin 2006; Giacintucci et al. 2009. Both radio halos
and relics show steep radio spectra (α≥1) and low surface brightness (∼ 10 −6 Jy arcsec −2 at
1.4 GHz).
The most widely accepted scenario for the origin of radio relics is that (primary) relativistic
electrons are injected into the ICM from AGN activity and/or from star formation in galaxies. They
are then accelerated by shocks or mergers (Enßlin et al. 1998; Röttiger, Burns & Stone 1999) or
by adiabatic compression of fossil radio plasma (Ensslin & Gopal-Krishna 2001). This picture is
supported by the typical high degree of polarization and the orientation of magnetic field vectors,
which appear to be oriented along the major axis of elongated relics with coherence scales of
several hundreds of kpc (Right panel of Fig. 3.1). Indeed, compression and/or shocks can locally
amplify and align the field with their surfaces.
In contrast, the formation of radio halos is still debated. Since primary relativistic electrons
lose energy on short timescales (∼ 10 7−8 yrs), they cannot diffuse any significant distance in
the cluster before ceasing to radiate. To explain the large extension, up to Mpc scales, of radio
halos, continuous injection processes and/or particle re-acceleration are required. These might
be due to gas turbulence, efficient in energetic mergers (primary models). Another possibility
is that secondary electrons are continuously injected in the ICM by hadronic collisions between
relativistic protons and ICM thermal protons (secondary models). Because of their higher mass,
23

24 3. Intergalactic magnetic fields
18:00.0
-23:20:00.0
Declination
22:00.0
24:00.0
26:00.0
28:00.0
500 kpc
50.0 40.0 20:03:30.0 20.0 10.0
Right ascension
Figure 3.2: Example of a radio halo. Chandra image of the hot ICM in the cluster RXCJ 2003.5–
2323 overlaid on contours of 235 MHz radio emission from the Giant Metrewave Radio Telescope
(GMRT). Taken from Giacintucci et al. (2009).
relativistic protons have loss timescales of the order of the Hubble time, thus they are able to travel
a large distance through the cluster.
The observed link between radio halos and relics and the unrelaxed state of clusters supports
the scenario of re-acceleration in situ of particles by turbulence produced during cluster mergers
e.g. Schlickeise, Sievers & Thiemann 1987; Brunetti et al. 2001; Petrosian 2001; Fujita, Takizawa
& Sarazin 2003).
Cluster mergers are indeed able to drive turbulence, shocks and compression in the intracluster
medium. The energy dissipated in cluster mergers can be channeled into the amplification
of magnetic fields and particle acceleration (e.g. Carilli & Taylor 2002; Dolag et al. 2002; Brüggen
et al. 2005; Brunetti & Lazarian 2007; Pfrommer, Enßlin & Springel 2008).
However, not all dynamically disturbed clusters possess radio halos. One of the reasons is
that they are difficult to observe, due to their low surface brightness and steep spectrum. Cassano
& Brunetti (2005) have shown that only the most massive clusters, where the energy density of the
turbulence is high enough, should possess radio halos produced by primary models.
3.1.2 Magnetic fields from diffuse radio sources
Secondary models predict gamma-ray emission from neutral pion decay from the same hadronic
collisions that create relativistic electrons. So far, such diffuse emission, which would corroborate
the secondary models for the origin of radio halos, has not been detected from clusters, and only
upper limits can be inferred (e.g. Ackermann et al. 2010). The expected gamma-ray flux is related
24

3.1. Introduction 25
to the relativistic electron energy spectrum, which together with the magnetic field strength also
determines the observed radio emission. Consequently, an upper limit to the gamma-ray flux
corresponds to a lower limit on the intra-cluster field strength. To reproduce the observed radio
halo emissivity, current gamma-ray upper limits give average field strength of the order of several
µG (e.g. Jeltema & Profumo 2011).
When halos or relics are observed, minimum energy assumptions offer an alternative approach
to estimate the magnetic field strength B eq averaged over the whole halo/relic volume. This
is the only estimate available for the preferred primary models. Multiple uncertainties affect
this approach (Sec. 1.3.4). In particular, equipartition calculations can grossly underestimate the
field strength as they are based on the assumption of a homogeneous magnetic field throughout
the halo/relic volume, in contrast with evidence for its radial decline (Sec. 3.5). With standard
assumptions (ζ= 1,Φ=1, emitting frequency range=10 MHz÷10 GHz) B eq ranges from 0.1 to
2µG in cluster halos and from 0.5 up to 6µG in relics (Ferrari et al. 2008 and reference therein,
van Weeren et al. 2010).
3.1.3 Diffuse inverse Compton emission
As in radio galaxies, the observation of diffuse IC flux from radio halos and relics would allow
a straightforward determination of both magnetic field strength and relativistic particle density
(Sec. 1.2.3). IC emission from radio halos and relics, should be observable at hard X-ray energies
(the “HXR excess”) where the exponential decline of the thermal bremsstrahlung, dominating at
soft keV energies, is steeper than the expected non-thermal spectrum (Rephaeli 1977).
So far, highly significant IC emission has been detected only in the Ophiuchus cluster
(e.g. Eckert et al. 2008), which possesses a mini-halo and the implied averaged magnetic field
is≈0.3µG (Murgia et al. 2010). Earlier, less significant, detections gave predicted magnetic fields
strength in the range 0.1-0.7µG, reaching the higher values in relics (e.g. Slee et al. 2001; Rephaeli,
Gruber & Arieli 2006; Eckert et al. 2008). Similar or slightly higher fields are derived as lower
limits from non-detections of IC emission (e.g. Lutovinov et al. 2008; Wik et al. 2011).
Estimates of magnetic fields from IC emission in clusters are problematic for a number of
reasons.
• It is difficult to distinguish the HXR excess from thermal emission. In some clusters, the
observed spectrum can instead be reproduced by thermal model with a single or multiple
gas temperatures (e.g. Lutovinov et al. 2008; Wik et al. 2011).
• An HXR excess might also be interpreted as synchrotron emission from highly relativistic
electrons (≈PeV, Timokhin et al. 2004) or non-thermal bremsstrahlung from supra-thermal
25

26 3. Intergalactic magnetic fields
electrons (Blasi 2000; Ensslin & Gopal-Krishna 2001). However, the latter scenario
is problematic because of the inefficiency of such process which converts most of the
collisional energy into heat (Petrosian 2001, 2003).
• They assume the coincidence of IC and synchrotron emitting particles, which is poorly
constrained if not unknown.
Taken at face value, IC detections predict slightly lower average field strengths than those
which assume equipartition between relativistic electrons and field in halos and relics. These
in turn are of about one order of magnitude below those derived from current upper limits on
gamma-ray emission for secondary models (any improvement in the sensitivity of gamma-ray
observations, will give even higher magnetic fields in this case). On the other hand, changes in
assumptions about the relative number densities of relativistic protons and electrons can lead to
increases in B eq by factors of 2–3 (e.g. Beck & Krause 2005). Because of the various uncertainties
and assumptions of all the methods outlined above, a detailed comparison of their results is not
sensible. We can just say that all of them agree thatµG average magnetic field strengths are
present in clusters.
The Faraday effect across radio galaxies discussed in the next sections provides a much more
detailed, and I would say surprising, picture of magnetic fields in the hot phase of the IGM.
3.2 Faraday Rotation
The Faraday effect (Faraday 1846) describes the rotation suffered by the plane of polarization
of linearly polarized radiation traveling through a magnetized thermal plasma. The presence
of magnetic fields induces different indices of refraction (circular birefringence) for the two
circularly polarized components (left versus right) into which linearly polarized radiation can be
decomposed. The different propagation speeds of the two components cause a shift between them
and consequently a rotation of the plane of polarization. Such a situation occurs for example when
polarized radio galaxies are located behind or embedded in magnetized intergalactic media.
The rotation∆Ψ of the E-vector position angle of linearly polarized radiation by a magnetized
thermal plasma is given by:
∆Ψ [rad] =Ψ(λ) [rad] −Ψ 0 [rad] =λ 2 [m 2 ] RM [rad m −2 ] , (3.1)
whereΨ(λ) andΨ 0 are the E-vector position angle of linearly polarized radiation observed at
wavelengthλ and the intrinsic angle, respectively. RM is the rotation measure. For a fully resolved
foreground Faraday screen, theλ 2 relation of Eq. 3.1 holds exactly at any observingλ.
26

3.2. Faraday Rotation 27
The RM can be expressed as:
∫ L[kpc]
RM [rad m −2 ] = 812 n e [cm −3 ]B z [µG] dz [kpc] , (3.2)
0
where n e is the electron gas density in the thermal plasma, B z is the magnetic field along the lineof-sight
and L is the integration path. Therefore the study of the Faraday effect across polarized
radio sources allows us to probe the magnetic field strength along the line-of-sight. As already
mentioned in Sec. 1.2.2, the polarization angle can be described by using the observables Stokes
parameters Q and U (Stokes 1852):
Ψ λ = 1 2 arctan (
Uλ
Q λ
)
, (3.3)
and it can be measured at several wavelengths by multi-frequency polarimetric radio observations.
Then, RM across radio sources can be derived through a linear fit of the observed polarization
angles as a function ofλ 2 (Eq. 3.1). As is well known, the determination of RM is complicated
because of the nπ ambiguities in the observedΨ obs .
Removal of these ambiguities requires
observations at least at three different wavelengths, well-spaced inλ 2 . An alternative is the method
of RM synthesis using simultaneous polarization observations over a contiguous frequency range,
which will be available with the new generation of wideband correlators (Brentjens & Bruyn
2005).
3.2.1 Internal Faraday Rotation
Internal RM occurs if thermal plasma and synchrotron emitting particles are mixed.
In this
case, together with the rotation of the polarization plane we observe a decrease of the degree of
polarization with increasing wavelength. Indeed, emission arising from different depths within a
source suffers differential Faraday rotation reducing the degree of polarization. The functional
form of the expected depolarization is in general related to the geometry of the source.
simplest modeling for such phenomenon is the optically-thin slab with uniform density and
magnetic field located entirely within the source. In this case, the degree of polarization is given
by (Burn 1966):
The
P Obs (λ)=P Int | sin(RM′ λ 2 )
RM ′ λ 2 |, (3.4)
where RM ′ is the internal Faraday RM and is equal to 1 2
of that expected for a foreground slab
screen of the same thermal density and field strength.
Even in the case of internal RM,λ 2 rotation holds at sufficiently short wavelengths.
particular, in the case of the slab model theλ 2 rotation is observed over 90 degrees then shows
27
In

450
400
350
300
300
250
200
150
350
300
250
200
300
250
200
150
350
300
250
200
250
200
150
100
28 3. Intergalactic magnetic fields
01
01
01
1100
01
01
01
0
01 00 1 1 0
01 0 0 1 1
01
01
1100
01
01
01
Figure 3.3: Examples of 01
plots of the E-vector position angleΨ
01
obs againstλ 2 at different locations
1100
in PKS 2149-158 The01
exact position of the points in the sources is shown in the insets of the
01
01
individual panels. The solid line represent the best fit of theλ 2 -law to the data. Taken from
Guidetti et al. (2008).
an abrupt 90 ◦ change. In more realistic geometries, it is the lack of significant deviations fromλ 2
1100
00 1 11 0
0
01
0 00 1 11 0
1100
1
1
01
01
rotation over≥45 ◦ which allows us to exclude the presence of internal RM. So far, there is little
evidence for RM occurring within the radio sources. This suggests that either that there is little
thermal plasma in radio galaxies or that the internal fields are tangled on small scales.
Fig. 3.3 shows some examples ofλ 2 rotation of polarization positions angle for the extended
radio galaxy PKS 2149-158 in A 2382 (Guidetti et al. 2008). This indicate a foreground origin for
most of the RM across the source. The results from Faraday rotation and depolarization analyses
presented in this thesis (Chapters 4, 5 and 6) are consistent with this scenario.
3.2.2 Foreground RM contributions
The RM occurring in front of radio galaxies can be thought of as the combined effect of multiple
intervening magneto-ionic media along the line-of-sight:
1. thermal gas in a skin local to the source;
2. the interstellar and intergalactic media surrounding the radio source;
3. the same components associated with other galaxies located along the line-of-sight;
4. the ISM of our Galaxy and
5. the Earth’s ionosphere.
A process able to form a magnetized local skin around the sources is the growth of Kevin-
Helmholtz instabilities at the surface of the radio lobes in contact with the surrounding medium
(Bicknell, Cameron & Gingold 1990). These surface waves are expected to cause advection of the
28

3.2. Faraday Rotation 29
Figure 3.4: RM (corrected for the Galactic contribution) plotted as a function of source impact
parameter in kpc for a sample of Abell clusters. The radio galaxies are split into clusters members
(red), background (blue) sources and sources whose line-of-sight does not intercept the clusters
X-ray emission (black). Taken from Clarke (2004).
magnetic field internal to the radio galaxies into the surrounding thermal gas with the consequent
creation of a local Faraday screen. This model predicts magnetic field reversals and depolarization
of emission from the surface layer.
Arguments in favour of an origin of the observed RM in the IGM of the radio source (item 2)
are as follows.
• The detection of diffuse synchrotron emission in galaxy clusters (Sec. 3.1.1), implies that
the IGM must be magnetized, so Faraday rotation is inevitable.
• The more distant radio lobe of a source shows higher depolarization than the approaching
one (the Laing-Garrington effect; Laing 1988; Garrington et al. 1988). This is interpreted
as resulting from a difference in the path length, and hence the Faraday depth, for the two
lobes. Such an effect is not expected for a skin model (item 1) since in this case there is no
reason for different path lengths between the two lobes.
• A statistical comparison of the RM’s of point sources behind, embedded in and away from
foreground galaxy clusters (Clarke 2004) has shown that those located behind clusters have
a large RM dispersion (Fig. 3.4).
• Very large RM values are observed in cool core clusters.
• The dispersion in RM observed across individual sources is not correlated with Galactic
coordinates, implying that the Galactic contribution is not dominant on small scales
(although it will influence the mean).
29

30 3. Intergalactic magnetic fields
• The presence of RM fluctuations on small angular scales also argues against a Galactic
origin.
The latter two points suggest a generally small Galactic contribution, which nonetheless can
become significant for radio sources located at low Galactic latitudes and when the Galactic
magnetic field is almost aligned along the line-of-sight. To derive the properties of the RM local
to the source, the Galactic contribution must be estimated and then properly removed; this is
discussed more in detail in the next section. Finally, the RM due to the Earth’s ionosphere at cm
wavelengths is expected to be at most a few degrees (less at solar minimum) and is generally small
compared with the fitting errors.
3.2.3 Galactic Faraday rotation
Our Galaxy has a magnetic field which can be thought as sum of an ordered component on kpc
scale and a random one fluctuating on smaller scales. Both the two fields have mean strength of
about 2µG e.g. (Beck 1996).
Most extragalactic radio sources show Faraday rotation measures of the order of 10 rad m −2
due to propagation of the emission through the magnetized interstellar medium of our Galaxy
(Simard-Normandin et al. 1981). Diffuse and ionized Galactic feature in front of the sources,
often associated with HII regions, represent another source of RM. At Galactic latitudes|b|≤5 ◦ ,
radio sources can show Galactic RM’s up to±300 rad m −2 . The Galactic RM can be estimated
by comparing the mean RM values of sources near the target, and assuming a smooth spatial
variation of the Galactic field. Dineen & Coles (2005) have derived spherical harmonic models
for the Galactic RM, by fitting to the RM values of large numbers of extragalactic sources.
There is evidence for linear gradients in Galactic RM on arcminute scales: Laing et al.
(2006) found a gradient of 0.025 rad m −2 arcsec −1 along the jets of the radio galaxy NGC 315
(l=124.6 ◦ , b=−32.5 ◦ ). They argued that this gradient is almost certainly Galactic in origin,
since the amplitude of the linear variation exceeds that of the small-scale fluctuations associated
with NGC 315 (see also the case of 3C 449 presented in Chapter 4). Although the estimate of
the Galactic RM might be highly uncertain in some cases, the angular size of radio galaxies are
usually small enough so that it can be reasonably approximated as constant across them. In this
case, the study of the RM local to the source is provided by statistical analysis techniques based
on the use of the structure function (Sec. 3.7.1) which is a powerful tool independent of the mean
level and (to first order) of structure on scales larger than the measurement area.
30

3.3. Depolarization 31
3.3 Depolarization
Faraday rotation generally leads to a decrease of the degree of polarization with increasing
wavelength, or depolarization (DP) in different circumstances. We define DP λ 1
λ 2
= p(λ 1 )/p(λ 2 ),
where p(λ) is the degree of polarization at a given wavelengthλ. We adopt the conventional
usage, in which higher depolarization corresponds to a lower value of DP.
Laing (1984) summarized the interpretation of polarization data. Faraday depolarization of
radio emission from radio sources can occur in three principal ways:
1. thermal plasma is mixed with the synchrotron emitting material (internal depolarization,
Sec. 3.2.1);
2. there are fluctuations of the foreground Faraday rotation across the beam (beam
depolarization) and
3. the polarization angle varies across the finite band of the receiving system (bandwidth
depolarization).
The last two terms are globally named as external depolarization and are due only to
limitations of the instrumental capabilities.
The beam depolarization is due to the presence of unresolved inhomogeneities of thermal
density and/or magnetic field which cause differential Faraday rotation of the polarization angle
within the observing beam, and in turn a decrease of the observed degree of polarization with
increasing wavelength. From a foreground Faraday screen with a small gradient of RM across the
beam, it is still possible to observeλ 2 rotation over a wide range of polarization angle. In this case,
the wavelength dependence of the depolarization is expected to follow the Burn law (Burn 1966)
p(λ)= p(0) exp(−kλ 4 ), (3.5)
where p(0) is the intrinsic value of the degree of polarization and k=2|∇RM| 2 σ 2 , with FWHM=
2σ(2 ln 2) 1/2 . Since k ∝ |∇RM| 2 , Eq. 3.5 clearly illustrates that higher RM gradients across
the beam generate higher k values and hence higher depolarization. The variation of p with
wavelength can potentially be used to estimate fluctuations of RM across the beam, which are
below the resolution limit. We can determine the intrinsic polarization p(0) and the proportionality
constant k by a linear fit to the logarithm of the observed fractional polarization as a function of
λ 4 .
To distinguish between the external and internal depolarization, sensitive polarization data at
multiple frequencies and resolutions are needed. A decrease in depolarization with increasing
31

32 3. Intergalactic magnetic fields
resolution, is expected from beam depolarization (Eq. 3.5). The external depolarization is
correlated with the amount of the RM gradient, while that occurring internally is expected to
be correlated with the geometry of the source and the RM values, in the sense that regions of small
RM correspond to low depolarization.
Finally, bandwidth depolarization occurs when a significant rotation of the polarization angle
of the radiation is produced across the finite bandwidth of the receiving system. The rotation of
the polarization angle across the observing band is:
∆Ψ=−2RMλ 2∆ν
ν
(3.6)
where∆ν is the bandwidth and ν is the central frequency. This will reduce the observed
polarization level by a quantity sin(∆Ψ)/∆Ψ below that for monochromatic radiation. None of
the observations used in this work are affected by significant bandwidth depolarization.
3.4 RM analysis
Observations of Faraday rotation variations across extended radio galaxies, once corrected for
the contribution from our Galaxy (Sec 3.2.3), allow us to derive information about the integral
of the density-weighted line-of-sight field component. The hot (T≃10 7 − 10 8 K) plasma emits
in the X-ray energy band via thermal bremsstrahlung (Sec. 2.2.1). When high quality X-ray data
for a radio-source environment are available, it is possible to infer the gas density distribution and
therefore to separate it from that of the magnetic field (Eq. 3.2), subject to some assumptions about
the relation of field strength and density. Consequently, uncertainties on these field estimates come
from the assumed magnetic field topology and gas density profile.
Faraday rotation measure analysis has a number of advantages over other methods for
estimating the magnetic field. RM studies can be carried out across radio galaxies hosted in
less dense environments, allowing the study of magnetic fields in systems too sparse for radio
halos to be detected. Moreover, the analysis of diffuse radio emission requires low-resolution and
low-frequency observations, making it difficult to derive detailed information on the structure of
the intergalactic magnetic field. These limitations do not affect RM maps, which can be produced
even at sub-arcsec angular resolution at high frequencies (≥5 GHz) as long as the radio galaxies
are bright enough in polarized intensity.
A detailed study of intergalactic magnetic fields through the analysis of diffuse radio sources
is also limited by the fact that these are affected by strong internal depolarization, intrinsically
related to the nature of the diffuse synchrotron emission itself. In contrast, so far there is little
evidence for internal depolarization in radio galaxies.
32

3.5. The magnetic field profile 33
Faraday rotation studies also provide information about the direction of the magnetic field,
being the positive (negative) when the magnetic field direction points towards (away from) the
observer. One potential problem is that there could be numerous field reversals along the line of
sight, which would be averaged out. The minimum field strength can be derived by assuming a
constant magnetic field along the line-of-sight. Such estimates generally give fields of about 1µG.
Finally, through an extrapolation of Eq. 3.1 to zero wavelength, RM studies allow us to
determine the intrinsic distribution of the projected magnetic field of radio galaxies, and its relation
to the environment.
3.5 The magnetic field profile
In order to estimate the magnetic field strength, the equipartition and IC analyses assume a constant
magnetic field through the whole halo or relic volume.
This assumption is an oversimplified
picture as is clear from a simple energy-balance argument: if the field was uniform on Mpc scales,
the magnetic pressure would exceed the thermal pressure in the outskirts of the clusters. Jaffe
(1980) first suggested that the magnetic field distribution in a cluster might be similar to those of
the thermal gas density and the volume density of massive galaxies and therefore would decline
with the cluster radius. Observations, analytical models and MHD simulations of galaxy clusters
all suggest that the magnetic field intensity should scale with the thermal gas density (e.g. Brunetti
et al. 2001; Dolag 2006; Dolag, Bykov & Diaferio 2008; Guidetti et al. 2008). Govoni et al.
(2001a) found a two-point spatial correlation between the X-ray and radio halo surface brightness
in the galaxy clusters of their sample suggesting that the thermal and non-thermal components
might have similar radial scalings. Another indication of a radial decrease of the magnetic field
strength comes from the radial steepening observed in a few radio halos (Coma, A665, A2163,
Giovannini et al. 1993; Feretti et al. 2004a) which are expected in the modeling of radio halos
formation including such a radial decrease.
Multiple works based on RM simulations (e.g. Murgia et al. 2004; Govoni et al. 2006; Guidetti
et al. 2008; Laing et al. 2008; Kuchar & Enßlin 2009; Bonafede et al. 2010) have considered a
radial field-strength variation of the form:
〈B 2 (r)〉 1/2 = B 0
[
ne (r)
n 0
] η
(3.7)
Here, B 0 is the rms magnetic field strength at the group/cluster centre and n e (r) is the thermal
electron gas density. The results from the simulations are in favour ofηin the range 0.5-1. This
functional form is consistent with other observations, analytical models and numerical simulations.
In particular,η=2/3 corresponds to flux-freezing andη=1/2 to equipartition between thermal
33

34 3. Intergalactic magnetic fields
and magnetic energy. Dolag et al. (2001, 2006) foundη≈1 from the correlation between the
observed rms RM and X-ray surface brightness in galaxy groups and clusters and showed that this
is consistent with the results of MHD simulations.
3.6 Tangled magnetic field
Most of the published RM images of radio galaxies located in different environments show random
structures with patches of different size, ranging from a few kpc up to tens of kpc (Fig. 3.5). From
these RM distributions it is clear that the magnetic fields are not regularly ordered on cluster (Mpc)
scales, but highly turbulent. It is indeed likely that anisotropies in the magnetic field would reflect
themselves in the RM images, since the projection connecting field and RM conserves anisotropy.
Faraday rotation by tangled intergalactic magnetic fields was considered in several early
theoretical papers (e.g. Lawler & Dennison 1982; Tribble 1991; Felten 1996). The simplest RM
modeling invokes a Faraday screen in which the magnetic field fluctuates on a single scaleΛ c . The
screen can be thought as made of cells of constant size and magnetic field strength, but random
magnetic field direction. The RM from such a screen will be produced by a random walk along the
line-of-sight. Because of the large number of cells, the RM distribution is expected to be described
by a Gaussian function with zero mean and dispersionσ RM given by:
∫
σ 2 RM =〈RM2 〉=812 2 Λ c (n e B z ) 2 dz. (3.8)
Considering a thermal density distribution which follows aβ-profile (Eq. 2.3), and an isotropic
field (so that B= √ 3B z ), Eq. 3.8 can be integrated analytically and gives (Felten 1996)
σ RM (r ⊥ )= KBn 0rc 1/2 Λ 1/2 √
c
(1+ r ⊥ Γ(3β−0.5)Γ(3β) (3.9)
rc
) (6β−1)/4
whereΛ c is the cell size, r ⊥ is the projected distance of from the cluster center,Γis the Gamma
function, and K is a factor which depends on the location of the radio source along the line-of-sight
(e.g. Carilli & Taylor 2002). The central field strength can be estimated by using Eq. 3.9:σ RM (r ⊥ )
can be measured from spatially resolved RM images and at a first level of approximationΛ c can
be assumed to be equal to the coherence length, deduced by-eye, of the RM map. It must however
borne in mind that the magnetic fields in this model are not divergence-free (Enßlin & Vogt 2003).
This method has been applied to both statistical samples (Clarke 2004) and to single objects,
e.g. Hydra A (Taylor & Perley 1993) 1993), A 119 (Feretti et al. 1999b), 3C 295 (Allen et al.
2001), A 514 (Govoni et al. 2001a), 3C 129 Taylor2001, A 400 and A 2634 (Eilek & Owen 2002).
The magnetic field strengths deduced from these analyses are in the range 4-20µG, assumingΛ c
34

3.6. Tangled magnetic field 35
rad m −2
RAD/M/M rad m −2 -150 -100 -50 0
-100 -50 0 50 100
-15 36 00
100 kpc
-15 38 00
-15 40 00
(a) (b) (c)
Figure 3.5: RM images of radio galaxies with tails: PKS 2149-158 and PKS 2149-158b (a), 3C 31
(b), and 3C 75 (c). Figures respectively taken from: Guidetti et al. (2008), Laing et al. (2008),
Eilek & Owen (2002).
of about 10 kpc, with the highest values found for radio galaxies in cooling-core clusters. None
of these magnetic fields are thought to be dynamically important, since they provide negligible
pressure compared with that of the thermal component.
Although RM images in general are not expected to show perfectly Gaussian distributions
because of the partial sampling of large spatial scales, the Gaussian function is found to be a
good representation for many RM distributions, supporting the scenario of a tangled and isotropic
magnetic field component along the line-of-sight.
However, in contrast with this scenario the
majority of the RM distributions show non-zero means〈RM〉 if averaged over areas of size
comparable with that of the radio source, even after removing the Galactic contribution. These
〈RM〉’s are thought to be due to large-scale magnetic field structure, whose fluctuations occur
on scales larger than those producing the RM dispersion. Therefore, the magnetic field cannot
be tangled only on a single scale, but it must have fluctuations over a wide range of spatial
scales: small-scale components are necessary to produce the smallest structures observed in the
RM images (Fig. 3.5) and structures on larger scales are needed to account for the non-zero RM
mean. For this reason, the investigation of the magnetic field power spectrum has been object
of many recent papers. Several studies (Enßlin & Vogt 2003; Enßlin & Vogt 2005; Murgia et al.
2004; Govoni et al. 2006; Guidetti et al. 2008; Laing et al. 2008; Kuchar & Enßlin 2009) have
shown that detailed RM images of radio galaxies can be used to infer not only the strength of the
cluster magnetic field, but also its power spectrum. All of these studies agree that random RM
structures can be accurately reproduced if the magnetic field is isotropic and randomly variable
with fluctuations on a wide range of spatial scales.
35

36 3. Intergalactic magnetic fields
3.7 The magnetic field power spectrum
In order to derive the three-dimensional magnetic field power spectrum from the analysis of the
RM images, it is necessary to assume statistical isotropy for the field, since only the component
of the magnetic field along the line-of-sight contributes to the observed RM. It is important
to underline that the turbulence in the intergalactic medium can be locally inhomogeneous, as
expected in the MHD regime, particularly regarding small scales fluctuations. However, it is
likely that the local anisotropies are isotropically distributed, so that whenever the volume sampled
is large enough these anisotropies tend to average out along any line-of-sight. Therefore, the
assumption of magnetic-field isotropy must be taken in the sense that the field has no preferred
direction when averaged over a sufficiently large volume.
If the intergalactic magnetic field can be approximated as Gaussian random variable, then
its spatial distribution can be described by the power spectrum of the component along the lineof-sight,
or its Fourier transform (the autocorrelation function, Wiener-Khinchin Theorem). For
intergalactic magnetic fields, these assumptions are justified by:
• the patchiness of most of the RM images across radio galaxies, consistent with isotropic
random magnetic fields
• the fact that a sum of a large number of independent and identically-distributed random
variables (i.e., the field components) approaches a Gaussian distribution (Central Limit
Theorem,Rice et al. 1955).
Following Laing et al. (2008), the three-dimensional magnetic field power spectrum can be
expressed as ŵ( f ) such that ŵ( f )d f x d f y d f z is the power in a volume d f x d f y d f z of frequency space,
with f (magnitude f ) representing a vector in the frequency space with the f z coordinate along the
line-of-sight. The other interesting property of the magnetic field is its auto-correlation length
which in term of ŵ( f ) is given by:
Λ B = 1 4
∫ ∞
0
∫ ∞
0
ŵ( f ) f d f
ŵ( f ) f 2 d f
(3.10)
The constant 1 4
differs by a factor of 4π from that in the equivalent expressions in Enßlin & Vogt
(2003, equation 39): a factor of 2π is due to the use of spatial frequency f rather than wave-number
k (k=2π f ) and a factor of 2 is due to the different integration limits. 1
Enßlin & Vogt (2003, 2005) have shown that typicallyΛ RM >Λ B since the former gives
more weight to the largest spatial scales. In the literature, these scales have often been assumed to
1 Enßlin & Vogt (2003) integrate from−∞ to+∞ in r and r ⊥ .
36

3.7. The magnetic field power spectrum 37
Figure 3.6: Comparison of theσ RM profiles obtained from simulations of a Gaussian and random
magnetic field (solid line) and the expectations from the analytical model of Eq. 3.9 by assuming
three differentΛ c :Λ min =6 kpc,Λ B =Λ Bz = 16 kpc,Λ RM =69 kpc (dashed and dotted lines). The
best agreement is given byΛ c =Λ B . The simulated magnetic field power spectrum has slope n=2.
The source is assumed to be halfway through the cluster and the magnetic field strength and gas
density have the same radial profiles in both the simulations and the analytical formulation. Taken
from Murgia et al. (2004).
be identical and the approximationΛ c =Λ RM has been used in the single-scale model (Eq. 3.9),
leading to underestimates of the magnetic field strength. Murgia et al. (2004) have shown thatΛ B
is a good approximation forΛ c in Eq. 3.9 and therefore that this length must be used to compare
the analytical predictions from the single-scale model with RM analyses assuming a multi-scale
field (Fig. 3.6). SinceΛ B depends on the underlying magnetic field power spectrum (Eq. 3.10), the
latter must be estimated first.
Multiple different estimators of spatial statistics of the RM distributions such as structure and
auto-correlation functions or a multi-scale statistic have been used to derive the field strength, its
relation to the gas density and its power spectrum (Murgia et al. 2004; Govoni et al. 2006; Guidetti
et al. 2008; Laing et al. 2008). The technique of Bayesian maximum likelihood has also been used
for this purpose (Enßlin & Vogt 2005; Kuchar & Enßlin 2009). These studies have shown that the
magnetic field power spectrum is well approximated by a power-law
ŵ( f )∝ f −q (3.11)
over a range of spatial frequencies, corresponding respectively to the outer and inner fluctuation
scales of the magnetic field 2 . Although the theory of intergalactic magnetic field fluctuations is
still debated, a power-law functional form is expected if the turbulence is mostly hydrodynamic,
2 Spatial frequency and scale are related in the sense: f= 1/Λ
37

38 3. Intergalactic magnetic fields
or in the MHD cascade investigated by Goldreich & Sridhar 1997.
The analysis of Vogt & Enßlin (2003, 2005) suggests that the magnetic field power spectrum
has a power law form with the slope appropriate for Kolmogorov turbulence and that the autocorrelation
length of the magnetic field fluctuations is a few kpc. Also Adaptive Mesh Refinement
(AMR) simulations by Brüggen et al. (2005) are consistent with the Kolmogorov slope. However,
the Kolmogorov theory (1941), which assumes homogeneous and incompressible turbulence,
cannot be rigorously applied because of the evidence in the intergalactic medium of gas radial
scaling and in some cases of shells of compressed gas. Moreover, the deduction of a Kolmogorov
slope could be premature: there is a degeneracy between the slope and the outer scale, which is
difficult to resolve with current Faraday rotation data (Murgia et al. 2004; Guidetti et al. 2008;
Laing et al. 2008). Indeed, Murgia et al. (2004) pointed out that shallower magnetic field power
spectra are possible if the magnetic field fluctuations have structure on scales of several tens of
kpc. Guidetti et al. (2008) showed that a power-law power spectrum with a Kolmogorov slope,
and an abrupt long-wavelength cut-off at 35 kpc gave a very good fit to their Faraday rotation
and depolarization data for the radio galaxies in A 2382, although a shallower slope extending to
longer wavelengths was not ruled out.
In the next section I will describe the method used in this thesis to derive the magnetic power
spectrum from our RM maps.
3.7.1 Magnetic field power spectrum from two-dimensional analysis
The relation between the magnetic field power spectrum and the observed RM distribution is in
general quite complicated, depending on the fluctuations in the thermal gas density, the geometry
of the source and the surrounding medium, and the effects of incomplete sampling. In order to
derive the magnetic field power spectrum, one must make the following assumptions (e.g. Guidetti
et al. 2008; Laing et al. 2008):
1. The observed Faraday rotation is due entirely to a foreground ionized medium. This can be
addressed by lack of deviation fromλ 2 rotation over a wide range of polarization position
angle and the lack of associated depolarization (Sec 3.2.1)
2. The magnetic field is an isotropic, Gaussian random variable
3. The form of the magnetic field power spectrum is independent of position
4. The magnetic field is distributed throughout the Faraday-rotating medium, whose density is
a smooth, spherically symmetric function
38

3.7. The magnetic field power spectrum 39
5. The amplitude of ŵ( f ) is spatially variable, but is a function only of the thermal electron
density.
These assumptions guarantee that the spatial distribution of the magnetic field can be described
entirely by its power spectrum ŵ( f ) and that for a medium of constant depth and density, the power
spectra of magnetic field and RM are proportional (Enßlin & Vogt 2003).
If the RM fluctuations are isotropic, the RM power spectrum Ĉ( f ⊥ ) is the Hankel transform
of the auto-correlation function C(r ⊥ ), defined as
C(r ⊥ )=〈RM(r ⊥ + r ′ ⊥)RM(r ′ ⊥)〉, (3.12)
where r ⊥ and r ′ ⊥ are vectors in the plane of the sky and〈〉 is an average over r′ ⊥ . One could
think of obtaining the magnetic field power spectrum directly by Fourier transforming Eq. 3.12.
In reality, the observations are affected first by the effects of convolution with the beam, which
modify the spatial statistics of RM, and secondly, by the limited size and irregular shape of the
sampling region (the region of the source over which the RM has been derived), which results in
a complicated window function (Enßlin & Vogt 2003) for computational work in the frequency
space. Finally, most of the useful properties of the auto-correlation function are related to the
outer scale at C(r ⊥ ) approaches the zero-level, which in most cases is uncertain in the presence of
large-scale fluctuations.
The alternative strategy proposed by Laing et al. (2008), and applied in the work of this
thesis, first estimates the power spectrum of the RM, Ĉ(f ⊥ ), where Ĉ(f ⊥ )d f x d f y is the power in
the area d f x d f y , and derives that of the three-dimensional magnetic-field ŵ(f). Laing et al. (2008)
demonstrated a procedure that takes into account the convolution effects and minimises the effects
of uncertainties in the zero-level. In particular, they showed that
1. in the short-wavelength limit (meaning that changes in Faraday rotation across the beam
are adequately represented as a linear gradient), the measured RM distribution is closely
approximated by the convolution of the true RM distribution with the observing beam
2. the structure function is a powerful and reliable statistical tool to quantify the two
dimensional fluctuations of RM, given that it is independent of the zero-level and structure
on scales larger than the area under investigation.
The structure function is defined by
S (r ⊥ )= (3.13)
39

40 3. Intergalactic magnetic fields
(Simonetti, Cordes & Spangler 1984; Minter & Spangler 1996). It is related to the autocorrelation
function C(r ⊥ ) for a sufficiently large averaging region by S (r ⊥ )=2[C(r ⊥ )−C(0)].
Laing et al. (2008) also derived the effects of convolution with the observing beam on the
observed structure function. For the special case of a power-law power spectrum (their Eq.
B2), they showed that the observed structure function after convolution can be heavily modified
even at separations up to a few times the FWHM of the observing beam. Fig. 3.7 illustrates the
effects of the convolution in two different power-law power spectra and corresponding structure
function. After convolution the two power spectra are indistinguishable. This effect must
be taken into account when comparing observed and predicted structure functions. However,
because the suppression of power on high spatial frequencies due to the convolution, this analysis
only constrains the power spectrum of the fluctuations on scales larger than the beam-width.
Complementary information can be derived from numerical simulations of depolarization with
fine spatial sampling (Laing et al. 2008; Guidetti et al. 2008) which constrain fluctuations of RM
below the resolution limit. indeed, the use of the structure function together with the Burn law k
represents a powerful technique to investigate the RM power spectrum over a wide range of spatial
scales (Laing et al. 2008).
In this thesis we have derived very detailed images of Faraday rotation and depolarization
across radio galaxies. The next Chapters will show that all of our observations are consistent with
pure foreground Faraday rotation. Where the RM structures can be reasonably approximated
as isotropic, the statistics of the magnetic-field fluctuations have been quantified by deriving
rotation measure structure functions, fitted using models derived from theoretical power spectra.
The minimum scale of the magnetic field variations have been derived from depolarization
measurements. However, I will also show that not all Faraday rotation maps are consistent with
an isotropic magnetic field. This forms the most important part of my thesis and describes an
unexpected phenomenon in the RM world.
40

3.7. The magnetic field power spectrum 41
Figure 3.7: (a), (c), (e): the two model RM power spectra discussed for the radio galaxy 3C 31 in
Laing et al. (2008). Solid line, broken power-law power spectrum with indices q high = 11/3, q low
= 2.32 and a break frequency of 0.062 arcsec −1 (their equation 8). Dashed line: power-law power
spectrum with q=2.39 and a high-frequency cut-off at f max = 0.144 arcsec −1 (their equation 9).
(b), (d), (f): structure functions computed for the power spectra in panels (a), (c) and (d), with the
same line codes. (a) and (b) no convolution; (c) and (d) 1.5 arcsec FWHM convolving beam; (e)
and (f) 5.5 arcsec FWHM convolving beam.
41

42 3. Intergalactic magnetic fields
42

Chapter 4
Structure of the magneto-ionic medium
around the Fanaroff-Riley Class I radio
galaxy 3C 449 ∗
M
agnetic fields associated with galaxy groups deserve to be investigated in more detail,
since their environments are more representative than those of rich clusters. Moreover,
observations, analytical models and MHD simulations of galaxy clusters all suggest that the
magnetic-field intensity should scale with the thermal gas density (e.g. Brunetti et al. 2001;
Dolag 2006; Guidetti et al. 2008; Bonafede et al. 2010). A key question is whether the relation
between magnetic field strength and density in galaxy groups is a continuation of this trend.
This Chapter presents a detailed analysis of Faraday rotation in 3C 449, a bright, extended radio
source hosted by the central galaxy of a nearby group. With the aim of shedding new light on
the environment around this source, I derive the statistical properties of the magnetic field from
observations of Faraday rotation, following the method developed by Murgia et al. (2004). I use
numerical simulations to predict the Faraday rotation for different strengths and power spectra of
the magnetic field.
With the assumed cosmology, at the distance of 3C 449 1 arcsec corresponds to 0.342 kpc.
4.1 The radio source 3C 449: general properties
I image and model the Faraday rotation distribution across the giant FR I radio source 3C 449,
whose environment is very similar to that of 3C 31. The optical counterpart of 3C 449,
UGC 12064, is a dumb-bell galaxy and is the most prominent member of the group of galaxies
∗ Guidetti et al. 2010, A&A, 514, 50
43

IMAGE: XMM_AS_1.3_OGEOM.FITS
CNTRS: 1.3_5.5_I_2000_XMM.FITS
0.5 1 1.5 2 2.5
44 4. The magneto-ionic medium around 3C 449
39 30 00
100 kpc
DECLINATION (J2000)
39 25 00
39 20 00
N Spur
N Jet
S Lobe
S Jet
N Lobe
S Spur
39 15 00
22 32 00
22 31 30 22 31 00
RIGHT ASCENSION (J2000)
22 30 30
Figure 4.1: Radio contours of 3C 449 at 1.365 GHz superposed on the XMM-Newton X-ray image
(courtesy of J. Croston, Croston et al. 2003). The radio contours start at 3σ I and increase by factors
of 2. The restoring beam is 5.5 arcsec FWHM. The main regions of 3C 449 discussed in the text
are labelled.
2231.2+3732 (Zwicky & Kowal 1968) The source is relatively nearby (z=0.017085, RC3.9, de
Vaucouleurs et al. 1991) and quite extended, both in angular (30 arcmin) and linear size, so it is an
ideal target for an analysis of the Faraday rotation distribution: detailed images can be constructed
that can serve as the basis of an accurate study of magnetic field power spectra.
The source 3C 449 was one of the first radio galaxies studied in detail with the VLA (Perley,
Willis & Scott 1979). High- and low-resolution radio data already exist and the source has been
mapped at many frequencies. The radio emission of 3C 449 (Fig. 4.1) is elongated in the N–S
direction and is characterized by long, two-sided jets with a striking mirror symmetry close to
the nucleus. The jets terminate in well-defined inner lobes, which fade into well polarized tails
(spurs), of which the southern one is more collimated. The spurs in turn expand to form diffuse
outer lobes.
The brightness ratio of the radio jets is very nearly 1, implying that they are close to the plane
of the sky if they are intrinsically symmetrical and have relativistic flow velocities similar to those
44

4.1. The radio source 3C 449: general properties 45
derived for other FR I jets (Perley, Willis & Scott 1979; Feretti et al. 1999a; Laing & Bridle 2002a).
Therefore the jets are assumed to lie exactly in the plane of the sky, which simplifies the geometry
of the Faraday-rotating medium.
Hot gas associated with the galaxy was detected on both the group and the galactic scales by
X-ray imaging (Hardcastle, Worrall & Birkinshaw, 1998; Croston et al. 2003). These observations
revealed deficits in the X-ray surface brightness at the positions of the outer radio lobes, suggesting
interactions with the surrounding material. Figure 4.1 shows radio contours at 1.365 GHz overlaid
on the X-ray emission as observed by the XMM-Newton satellite (Croston et al. 2003). The X-ray
radial surface brightness profile of 3C 449 derived from these data can be fitted with the sum of a
point-source convolved with the instrumental response and aβmodel (Cavaliere & Fusco-Femiano
1976):
n e (r)=n 0 (1+r 2 /rc) 2 − 3 2 β , (4.1)
where r, r c and n 0 are the distance from the group X-ray center, the group core radius, and
the central electron density, respectively. Croston et al. (2008) found a best fitting model with
β=0.42±0.05, r c = 57.1 arcsec and n 0 = 3.7×10 −3 cm −3 . In these calculations, I assumed that
the group gas density is described by the model of Croston et al. (2008). The X-ray depressions
noted by Croston et al. (2003) are at distances larger than those at which it is possible to detect
linear polarization, and there is no direct evidence of smaller cavities close to the nucleus. I
therefore neglect any departures from the spherically-symmetrical density model, noting that this
approximation may become increasingly inaccurate where the source widens (i.e. in the inner
lobes and spurs).
The source 3C 449 resembles 3C 31 in environment and in radio morphology: both sources are
associated with the central members of groups of galaxies, and their redshifts are very similar. The
nearest neighbours are at a projected distances of about 30 kpc in both cases. Both radio sources
have large angular extents, bending jets and long, narrow tails with low surface brightnesses and
steep spectra, although 3C 31 appears much more distorted on large scales. There is one significant
difference: the inner jets of 3C 31 are thought to be inclined by≈50 ◦ to the line-of-sight (Laing
& Bridle 2002a), whereas those in 3C 449 are likely to be close to the plane of the sky (Feretti et
al. 1999a). It is therefore expected that the magnetized foreground medium will be very similar
in the two sources, but that the geometry will be significantly different, leading to a much more
symmetrical distribution of Faraday rotation in 3C 449 compared with that observed in 3C 31 by
Laing et al. (2008).
45

46 4. The magneto-ionic medium around 3C 449
4.2 Total intensity and polarization properties
The VLA observations and their reduction were presented by Feretti et al. (1999a). The high
quality of these data makes this source suited for a very detailed analysis of the statistics of the
Faraday rotation.
I produced total intensity (I) and polarization (Q and U) images at frequencies in the range
1.365 – 8.385 GHz from the combined, self-calibrated u-v datasets described by Feretti et al.
(1999a). The center frequencies and bandwidths are listed in Table 4.1. Each frequency channel
was imaged separately, except for those at 8.245 and 8.445 GHz, which were averaged. The
analysis below confirms that these frequency-bandwidth combinations lead to negligible Faraday
rotation across the channels, as already noted by Feretti et al. (1999a). All of the datasets were
imaged with Gaussian tapering in the u-v plane to give resolutions of 1.25 arcsec and 5.5 arcsec
FWHM,CLEANed and restored with circular Gaussian beams. The first angular resolution is
the highest possible at all frequencies and provides good signal-to-noise for the radio emission
within 150 arcsec (≃50 kpc) of the radio core (the well defined radio jets and the inner lobes),
while minimizing beam depolarization. The lower resolution of 5.5 arcsec allows imaging of the
extended emission as far as 300 arcsec (≃100 kpc) from the core at frequencies from 1.365 –
4.985 GHz (the 8.385-GHz dataset does not have adequate sensitivity to image the outer parts of
the source). Therefore it is possible to study the structure of the magnetic field in the spur regions,
which lie well outside the bulk of the X-ray emitting gas. Noise levels for both sets of images are
given in Table 4.1. Note that the maximum scales of structure, which can be imaged reliably with
the VLA at 8.4 and 5 GHz are≈180 and≈300 arcsec, respectively (Ulvestad, Perley & Chandler
2009). For this reason, I only use the Stokes I images for quantitative analysis within half these
distances of the core. The Q and U images have much less structure on these large scales and are
reliable to distances of±150 arcsec at 8.4 GHz and±300 arcsec at 5 GHz, limited by sensitivity
rather than systematic errors due to missing flux as in the case of the I image.
Images of polarized intensity P=(Q 2 + U 2 ) 1/2 (corrected for Ricean bias, following Wardle
& Kronberg (1974), fractional polarization p= P/I and polarization angleΨ=(1/2) arctan(U/Q)
were derived from the I, Q, and U images.
All of the polarization images (P, p,Ψ) at a given frequency were blanked where the rms
error inΨ > 10 ◦ at any frequency. I then calculated the scalar mean degree of polarization
〈p〉 for each frequency and resolution; the results are listed in Table 4.1. The values of〈p〉 are
higher at 5.5 arcsec resolution than at 1.25 arcsec because of the contribution of the extended
and highly polarized emission, which is not seen at the higher resolution. At 1.25 arcsec, where
the beam depolarization is minimized, the mean fractional polarization shows a steady increase
46

4.3. The Faraday rotation in 3C 449 47
Table 4.1: Parameters of the total intensity and polarization images.
ν Bandwidth 1.25 arcsec 5.5 arcsec
σ I σ QU 〈p〉 σ I σ QU 〈p〉
(GHz) (MHz) (mJy/beam) (mJy/beam) (mJy/beam) (mJy/beam)
1.365 12.5 0.037 0.030 0.24 0.018 0.014 0.26
1.445 12.5 0.021 0.020 0.25 0.020 0.011 0.27
1.465 12.5 0.048 0.049 0.25 0.019 0.013 0.25
1.485 12.5 0.035 0.027 0.21 0.014 0.010 0.26
4.685 50.0 0.017 0.017 0.32 0.017 0.013 0.37
4.985 50.0 0.018 0.017 0.33 0.017 0.016 0.39
8.385 100.0 0.014 0.011 0.31 0.015 0.013 −
Col. 1: Observation frequency. Col. 2: Bandwidth (note that the images at 8.385 GHz are derived from the average of two frequency
channels, both with bandwidths of 50 MHz, centered on 8.285 and 8.485 GHz); Cols.3, 4 : rms noise levels in total intensity (σ I )
and linear polarization (σ QU , the average ofσ Q andσ U ) at 1.25 arcsec FWHM resolution; Col. 5: mean degree of polarization at
1.25 arcsec; Col. 6, 7: rms noise levels for the 5.5 arcsec images; Col. 8: mean degree of polarization at 5.5 arcsec. I estimate that
the uncertainty in the degree of polarization, which is dominated by systematic deconvolution errors on the I images, is≈0.02
at each frequency.
from 1.365 to 4.685 GHz, where it reaches an average value of 0.32 and then remains roughly
constant at higher frequencies, suggesting that the depolarization between 4.685 and 8.385 GHz is
insignificant.
4.3 The Faraday rotation in 3C 449
4.3.1 Rotation measure images
I produced images of RM and its associated rms error by weighted least-squares fitting to the
polarization angle maps (Eq. 3.1) with resolutions of 1.25 arcsec and 5.5 arcsec (Fig. 4.2a and b)
using a version of theAIPS taskRM modified by G. B. Taylor. The 1.25 arcsec-RM map was made
by combining the maps of the polarization E-vector (Ψ) at all the seven available frequencies, so
that the sampling ofλ 2 is very good. The RM map was calculated with a weighted least-squares fit
at pixels with polarization angle uncertainties

48 4. The magneto-ionic medium around 3C 449
IMAGE: SUBIM.FITS
IMAGE: SUBIM.FITS
;220 ;200 ;180 ;160 ;140 ;120 ;100
RAD/M/M
;200 ;150 ;100
RAD/M/M
39 24 00
39 27 00
10 kpc
10 kpc
39 23 00
N Lobe
N Spur
39 24 00
DECLINATION (J2000)
39 22 00
39 21 00
N Jet
S Jet
DECLINATION (J2000)
39 21 00
(c)
39 20 00
S Lobe
39 18 00
S Spur
39 19 00
(a)
22 31 24 22 31 21 22 31 18
RIGHT ASCENSION (J2000)
(b)
22 31 30 22 31 24 22 31 18 22 31 12
RIGHT ASCENSION (J2000)
(d)
Figure 4.2: (a): Image of the rotation measure of 3C 449 at a resolution of 1.25 arcsec FWHM,
computed at the seven frequencies between 1.365 and 8.385 GHz. (b): Image of the rotation
measure of 3C 449 at a resolution of 5.5 arcsec FWHM, computed at the six frequencies between
1.365 and 4.985 GHz. In both of the RM images, the sub-regions used for the two-dimensional
analysis of Sec. 4.5 are labelled. (c) and (d): profiles ofσ RM as a function of the projected
distance from the radio source center. The points represent the values ofσ RM evaluated in boxes
as described in the text. The horizontal and vertical bars represent the bin widths and the rms on
the mean expected from fitting errors, respectively. Positive distances are in the direction of the
north jet and the vertical dashed lines show the position of the nucleus.
dominated by the Galactic contribution (see Sec. 4.3.2). The RM distribution peaks at
−161.7 rad m −2 , with a rms dispersion σ RM =19.7 rad m −2 . Note that I have not corrected
the values ofσ RM for the fitting errorσ RMfit . A first order correction would beσ RMtrue =
(σ 2 RM −σ2 RM fit
) 1/2 . Given the low value forσ RMfit , the effect of this correction would be very
small.
As was noted by Feretti et al. (1999a), the RM distribution in the inner jets is highly symmetric
about the core with RM≃ −197 rad m −2 at distances< ∼ 15 arcsec. The symmetry of the RM
distribution in the jets is broken at larger distances from the core: while the RM structure in
the southern jet is homogeneous, with values around −130 rad m −2 , fluctuations on scales of
≃10 arcsec (≃ 3 kpc) around a〈RM〉 of −160 rad m −2 are present in the northern jet. The lobes
are characterized by similar patchy RM structures with mean values〈RM〉≃−164 rad m −2 and
48

4.3. The Faraday rotation in 3C 449 49
Figure 4.3: Plots of E-vector position
angleΨagainstλ 2 at representative
points of the 5.5-arcsec RM map. Fits
to the relationΨ(λ)=Ψ 0 + RMλ 2 are
shown. The values of RM are given
in the individual panels.
σ RM ≃16 rad m −2 .
At 5.5 arcsec resolution, more extended polarized regions of 3C 449 can be mapped with
good sampling inλ 2 . The average fitting error is≃1.0 rad m −2 . Both spurs are characterized by
〈RM〉≃−160 rad m −2 , withσ RM =15 and 10 rad m −2 in the north and south, respectively. The
overall mean and rms for the 5.5 arcsec image,〈RM〉=−160.7 rad m −2 andσ RM =18.9 rad m −2 ,
are very close to those determined at higher resolution and are consistent with the integrated value
of−162±1 rad m −2 derived by Simard-Normandin et al. (1981).
It was demonstrated by Feretti et al. (1999a) that the polarization position angles at 1.25 arcsec
resolution accurately follow the relation∆Ψ ∝ λ 2 over a wide range of rotation. I find the
same effect at lower resolution: plots of E-vector position angleΨagainstλ 2 at representative
points of the 5.5 arcsec-RM image are shown in Fig 4.3. As at the higher resolution, there are no
significant deviations from the relation∆Ψ∝λ 2 over a range of rotation∆Ψ of 600 ◦ , confirming
that a foreground magnetized medium is responsible for the majority of the Faraday rotation and
extending this result to regions of lower surface brightness.
49

50 4. The magneto-ionic medium around 3C 449
In Fig. 4.2(c) and (d), I show profiles ofσ RM for both low and high resolution RM images. The
1.25 arcsec profile was obtained by averaging over boxes with lengths ranging from 9 to 13 kpc
along the radio axis; for the 5.5 arcsec profile I used boxes with a fixed length of 9 kpc (these sizes
were chosen to give an adequate number of independent points per box). The boxes extend far
enough perpendicular to the source axis to include all unblanked pixels. In both plots, there is
clear evidence for a decrease in the observedσ RM towards the periphery of the source, the value
dropping from≃30 rad m −2 close to the nucleus to≃10 rad m −2 at 50 kpc. This is qualitatively
as expected for foreground Faraday rotation by a medium whose density (and presumably also
magnetic field strength) decreases with radius. The symmetry of theσ RM profiles is consistent
with the assumption that the radio source lies in the plane of the sky.
4.3.2 The Galactic Faraday rotation
For the purpose of this work, 3C 449 has an unfortunate line-of-sight within our Galaxy. Firstly, the
source is located at l=95.4 ◦ , b=−15.9 ◦ in Galactic coordinates, where the Galactic magnetic
field is known to be aligned almost along the line-of-sight. Secondly, there is evidence from radio
and optical imaging for a diffuse, ionized Galactic feature in front of 3C 449, perhaps associated
with the nearby HII region S126 (Andernach et al. 1992). Estimates of the Galactic foreground
RM at the position of 3C 449 from observations of other radio sources are uncertain: Andernach et
al. (1992) found a mean value of−212 rad m −2 for six nearby sources, but the spherical harmonic
models of Dineen & Coles (2005), which are derived by fitting to the RM values of large numbers
of extragalactic sources, predict−135 rad m −2 . Nevertheless, it is clear that the bulk of the mean
RM of 3C 449 must be Galactic.
In order to investigate the magnetized plasma local to 3C 449, the value and possible spatial
variation of this Galactic contribution must be constrained. The profiles ofσ RM (Fig. 4.2) show
that the small-scale fluctuations of RM drop rapidly with distance from the nucleus. I might
therefore expect the Galactic contribution to dominate on the largest scales. At low resolution, the
RM can be accurately determined out to≈100 kpc from the core. This is roughly 5 core radii for
the X-ray emission and therefore well outside the bulk of the intra-group gas.
In order to estimate the Galactic RM contribution, I averaged the 5.5-arcsec RM image in
boxes of length 20 kpc along the radio axis (the box size has been increased from that of Fig. 4.2
to improve the display of large-scale variations). The profile of〈RM〉 against the distance from
the radio core is shown in Fig. 4.4. The large deviations from the mean in the innermost two bins
are associated with the maximum inσ RM and are almost certainly due to the intra-group medium.
The dispersion in〈RM〉 is quite small in the south and the value of〈RM〉=−160.7 rad m −2 for the
whole source is very close to that of the outer south jet. There are significant fluctuations in the
50

4.4. Depolarization 51
Figure 4.4: Profile of RM averaged over
boxes of length 20 kpc along the radio axis
for the 5.5 arcsec image. The horizontal bars
represent the bin width. The vertical bars
are the errors on the mean calculated from
the dispersion in the boxes, the contribution
from the fitting error is negligible and is not
taken into account. Positive distances are
in the direction of the north jet. The black
vertical dashed line indicates the position of
the nucleus; the green dashed line shows the
adopted mean value for the Galactic RM.
north, however. Given their rather small scale (∼300 arcsec), it is most likely that these arise in the
local environment of 3C 449, and I include them in the statistical analysis given below.
There is some evidence for linear gradients in Galactic RM on arcminute scales (Laing et
al. 2006a). In order to check the effect of a large-scale Galactic RM gradient on these results, I
computed an unweighted least-squares fit of a function〈RM〉= RM 0 + ax, where a and RM 0 are
constant and x is measured along the radio axis. The two innermost bins in Fig. 4.4 were excluded
from the fit. The best estimate for the gradient is very small: a=0.0054 rad m −2 arcsec −1 . I have
verified that subtraction of this gradient has a negligible effect on the structure-function analysis
given in Sec. 4.6.3.
I therefore adopt a constant value of−160.7 rad m −2 as the Galactic contribution.
4.4 Depolarization
I first estimated the bandwidth effects on the polarized emission of 3C 449 using the RM
measurements from Sec. 4.3.1. In the worst case (the highest absolute RM value of−240 rad m −2 )
at the lowest frequency of 1.365 GHz) the rotation across the band is≈10 ◦ . This results in a
depolarization of 0.017, negligible compared with errors due to noise.
The analysis of the depolarization of 3C 449 is based on the approach of Laing et al. (2008).
I made images of k at both standard resolutions by weighted least-squares fitting to the
fractional polarization maps (Eq. 3.5), using theFARADAY code by M. Murgia. The same frequencies
were used as for the RM images: 8.385 – 1.365 GHz and 4.985 – 1.365 GHz at 1.25 and 5.5 arcsec
resolution, respectively. By simulating the error distributions for p, I established that the mean
values of k were biased significantly at low signal-to-noise (cf. Laing et al. 2008), so only data
51

52 4. The magneto-ionic medium around 3C 449
IMAGE: K;BURN_4P_2000.FITS
IMAGE: K;BURN_5.5.FITS
;400 ;200 0 200 400 600
;400 ;200 0 200 400 600 800 1000
39 24 00
39 27 00
39 23 00
39 24 00
DECLINATION (J2000)
39 22 00
39 21 00
DECLINATION (J2000)
39 21 00
(c)
39 20 00
39 18 00
39 19 00
(a)
22 31 24 22 31 21 22 31 18
RIGHT ASCENSION (J2000)
(b)
22 31 30 22 31 24 22 31 18 22 31 12
RIGHT ASCENSION (J2000)
(d)
Figure 4.5: (a): image of the Burn law k in rad 2 m −4 computed from a fit to the relation
p(λ) = p(0) exp(−kλ 4 ) for seven frequencies between 1.365 and 8.385 GHz. (b): as (a), but
the angular resolution is 5.5 arcsec FWHM, and the k image has been computed computed from
the fit to the six frequencies between 1.365 and 4.985 GHz. (c) and (d): profiles for k as functions
of the projected distance from the radio source center (boxes as in Fig. 4.2). The horizontal and
vertical bars represent the bin widths and the error on the mean, respectively. Positive distances
are in the direction of the north jet, and the vertical dashed lines show the position of the nucleus.
with p>4σ p at each frequency are included in the fits. I estimate that any bias is negligible
compared with the fitting error. I also derived profiles of k with the same sets of boxes as for the
σ RM profiles in Fig. 4.2.
The 1.25 arcsec resolution k map is shown in Fig. 4.5(a), together with the profile of the k
values (Fig. 4.5c). The fit to aλ 4 law is very good everywhere: examples of fits at selected pixels
in the jets and lobes are shown in in Fig. 4.6. The symmetry observed in theσ RM profiles is
also seen in the 1.25 arcsec k image (Fig. 4.5): the mean values of k are≃50 rad 2 m −4 for both
lobes, 107 and 82 rad 2 m −4 for the northern and southern jet, respectively. The region with the
highest depolarization is in the northern jet, very close to the core and along the west side. The
integrated value of k at this resolution is≃56 rad 2 m −4 , corresponding to a mean depolarization
DP 20cm
3cm ≃ 0.87.
The image and profile of k at 5.5-arcsec resolution are shown in Fig. 4.5(b) and (d). The fit
52

4.4. Depolarization 53
1.25 arcsec 5.5 arcsec
Figure 4.6: Plots of degree of polarization, p (log scale) againstλ 4 for representative points at
1.25-arcsec and 5.5-arcsec resolution. Burn law fits (Eq. 3.5) are also plotted. The values of k are
quoted in the individual panels.
to aλ 4 law is in general good and examples are shown in in Figs. 4.6. As mentioned earlier, the
maximum scale of structure imaged accurately in total intensity at 5 GHz is∼300 arcsec (100 kpc)
and there are likely to be significant systematic errors in the degree of polarization on larger scales.
I therefore show the profile only for the inner±50 kpc. Over this range, the k profiles are quite
symmetrical, as at higher resolution. Note also that the small regions of very high k at the edge of
the northern and southern spurs in the map shown in Fig. 4.5 are likely to be spurious.
The mean values of k are≃184 rad 2 m −4 and 178 rad 2 m −4 for the northern and southern lobes,
respectively; and≃238 and 174 rad 2 m −4 in the northern and in the southern spurs. The integrated
value of k is≃194 rad 2 m −4 , corresponding to a depolarization DP 20cm
6cm ≃ 0.64.
To summarize, depolarization between 20 cm and 3 cm is observed. Since I measure lower
values of k at 1.25 arcsec than 5.5 arcsec, there is less depolarization at high resolution, as expected
for beam depolarization. The highest depolarization is observed in a region of the northern jet,
close to the radio core and associated with a steep RM gradient. Depolarization is significantly
higher close to the nucleus, which is consistent with the higher path length through the group gas
observed in X-rays. Aside from this global variation, I found no evidence for a detailed correlation
of depolarization with source structure. Depolarization and RM data are therefore both consistent
with a foreground Faraday screen. In Sec. 4.5.2 I show that the residual depolarization at 1.25-
arcsec resolution can be produced by RM fluctuations on scales smaller than the beam-width, but
higher-resolution observations are needed to establish this conclusively.
53

54 4. The magneto-ionic medium around 3C 449
4.5 Two dimensional analysis
4.5.1 General considerations
In order to interpret the fluctuations of the magnetic field responsible for the observed RM and
depolarization of 3C 449, I first discuss the statistics of the RM fluctuations in two dimensions. I
use the notation of Laing et al. (2008) in which f= ( f x , f y , f z ) is a vector in the spatial frequency
domain, corresponding to the position vector r=(x, y, z). The z-axis is taken to be along the
line-of-sight, so that the vector r ⊥ = (x, y) is in the plane of the sky and f ⊥ = ( f x , f y ) is the
corresponding spatial frequency vector. The goal is to estimate the RM power spectrum Ĉ(f ⊥ ),
where Ĉ(f ⊥ )d f x d f y is the power in the area d f x d f y and in turn to derive the three-dimensional
magnetic-field power spectrum ŵ(f), defined so that ŵ(f)d f x d f y d f z is the power in a volume
d f x d f y d f z of frequency space.
In order to derive the magnetic-field power spectrum, I make the simplifying assumptions
discussed in Sec. 3.7.1, which can be summarized as follows: the observed Faraday rotation occurs
in a foreground screen (in agreement with the results in Secs. 4.3 and 4.4) whose density is a
spherically symmetric function; the magnetic field is distributed throughout the Faraday-rotating
medium and it is an isotropic, Gaussian random variable, whose power spectrum is independent of
position; the power spectrum amplitude is a function only of the thermal electron density. These
assumptions guarantee that the spatial distribution of the magnetic field can be described entirely
by its power spectrum ŵ( f ) and that for a medium of constant depth and density, the power spectra
of magnetic field and RM are proportional (Enßlin & Vogt 2003)
In Sec. 4.3.2, I showed that the Galactic contribution to the 3C 449 RM is substantial and
argued that a constant value of−160.7 rad m −2 is the best estimate for its value. Fluctuations
in the Galactic magnetic field on scales comparable with the size of the radio sources could be
present; conversely, the local environment of the source might make a significant contribution
to the mean RM. Both of these possibilities lead to difficulties in the use of the autocorrelation
function. Therefore, following the approach of Laing et al. (2008), I initially used the RM
structure function (Eq. 3.13) to determine the form of field power spectrum (Sec. 4.5.2), while
for its normalization (determined by global variations of density and magnetic field strength), I
made use of three-dimensional simulations (Sec. 4.6).
4.5.2 Structure functions
I calculated the structure function for discrete regions of 3C 449, over which the spatial variations
of thermal gas density, rms magnetic field strength and path length are expected to be reasonably
54

4.5. Two dimensional analysis 55
small. For each of these regions, I first made unweighted fits of model structure functions derived
from power spectra with simple, parametrized functional forms, accounting for convolution with
the observing beam. I then generated multiple realizations of a Gaussian, isotropic, random RM
field, with the best-fitting power spectrum on the observed grids, again taking into account the
effects of the convolving beam. Finally, I made a weighted fit using the dispersion of the synthetic
structure functions as estimates of the statistical errors for the observed structure functions, which
are impossible to quantify analytically (Laing et al. 2008).
These errors, which result from
incomplete sampling, are much larger than those due to noise, but depend only weakly on the
precise form of the underlying power spectrum. The measure of the goodness of fit isχ 2 , summed
over a range of separations from r ⊥ = FWHM to roughly half of the size of the region: there is no
information in the structure function for scales smaller than the beam, and the upper limit is set
by sampling. The errors are, of course, much higher at the large spatial scales, which are less well
sampled. Note, however, that estimates of the structure function from neighbouring bins are not
statistically independent, so it is not straightforward to define the effective number of degrees of
freedom.
I selected six regions for the structure-function analysis, as shown in Fig. 4.2. These are
symmetrically placed about the nucleus, consistent with the orientation of the radio jets close to
the plane of the sky. For the north and south jets, I derived the structure functions only at 1.25-
arcsec resolution, as the low-resolution RM image shows no additional structure and has poorer
sampling. For the north and south lobes, I computed the structure functions at both resolutions
over identical areas and compared them. The agreement is very good, and the low-resolution RM
images do not sample significantly larger spatial scales, so I show only the 1.25-arcsec results.
Finally, I used the 5.5-arcsec RM images to compute the structure functions for the north and
south spurs, which are not detected at the higher resolution.
The structure function has a positive bias given by 2σ 2 noise , whereσ noise is the uncorrelated
random noise in the RM image (Simonetti, Cordes & Spangler 1984). The mean noise of the
1.25 and 5.5-arcsec RM maps is

56 4. The magneto-ionic medium around 3C 449
the structure function, including convolution (Laing et al. 2008), and therefore to avoid numerical
integration.
The fits were quite good, but systematically gave slightly too much power on small spatial
scales and over-predicted the depolarization. I therefore fit a cut-off power law (CPL) power
spectrum of the form
Ĉ( f ⊥ ) = 0 f ⊥ < f min
= C 0 f −q
⊥
f ⊥ ≤ f max
= 0 f ⊥ > f max . (4.3)
Initially, I consider values of f min sufficiently small that their effects on the structure functions
over the observed range of separations are negligible. The free parameters of the fit in this case
are the slope, q, the cut-off spatial frequency f max and the normalization of the power spectrum,
C 0 . In Table 4.2, I give the best-fitting parameters for CPL fits to all of the individual regions. The
fitted model structure functions are plotted in Fig. 4.7(a)–(f), together with error bars derived from
multiple realizations of the power spectrum as in Laing et al. (2008).
In order to constrain RM structure on spatial scales below the beam-width, I estimated the
depolarization expected from the best power spectrum for each of the regions with 1.25-arcsec
RM images, following the approach of Laing et al. (2008). To do this, multiple realizations of
RM images have been made on an 8192 2 grid with fine spatial sampling. I then derived the Q
and U images at our observing frequencies, convolved to the appropriate resolution and compared
the predicted and observed mean degrees of polarization. These values are given in Table 4.4.
The uncertainties in the expected< k > in Table 4.4 represent statistical errors determined
from multiple realizations of RM images with the same set of power spectrum parameters. The
predicted and observed values are in excellent agreement. A constant value of f max = 1.67 arcsec −1
predicts very similar values, also listed in Table 4.4. I have not compared the depolarization data
at 5.5-arcsec resolution in the spurs because of limited coverage of large spatial scales in the I
images (Sec. 4.2), which is likely to introduce systematic errors at 4.6 and 5.0 GHz.
I performed a joint fit of the CPL power spectra, minimizing theχ 2 summed over all six subregions,
giving equal weight to each and allowing the normalizations to vary independently. In this
case the free parameters of the fit are the six normalizations (one for each sub-region), the slope,
and the maximum spatial frequency. The joint best-fitting single power-law power spectrum has
q=2.68.
A single power law slope does not give a good fit to all of the regions simultaneously, however.
It is clear from Fig. 4.7 and Table 4.2 that there is a flattening in the slope of the observed structure
56

4.5. Two dimensional analysis 57
Figure 4.7: (a)-(f): Plots of the RM structure functions for the sub-regions showed in Fig.4.2.
The horizontal bars represent the bin widths and the crosses the centroids for data included in the
bins. The red lines are the predictions for the CPL power spectra described in the text, including
the effects of the convolving beam. The vertical error bars are the rms variations for the structure
functions derived using a CPL power spectrum with the quoted value of q on the observed grid of
points for each sub-region. (g)-(l): as (a)-(f), but using a BPL power spectra with fixed slopes and
break frequency, but variable normalization.
functions on the largest scales (which are sampled primarily by the spurs). In order to fit all of the
data accurately with a single functional form for the power spectrum, I adopt a broken power law
form (BPL) for the RM power spectrum:
Ĉ( f ⊥ ) = 0 f ⊥ < f min
= D 0 f (q l−q h )
b
f −q l
⊥
= D 0 f −q h
⊥
f b ≥ f ⊥
f max ≥ f ⊥ > f b
= 0 f ⊥ > f max . (4.4)
I performed a BPL joint fit in the same way as for the CPL power spectra.
In this case
57

58 4. The magneto-ionic medium around 3C 449
Table 4.2: CPL power spectrum parameters for the six individual sub-regions of 3C 449.
Region FWHM CPL
(arcsec ) Best Fit Min Slope Max Slope
q f max q − f max q + f max
N SPUR 5.50 2.53 1.96 1.58 0.23 3.44 ∞
N LOBE 1.25 2.87 1.60 2.31 0.55 3.35 ∞
N JET 1.25 3.15 1.21 2.29 0.30 4.27 ∞
S JET 1.25 2.76 1.95 2.02 0.3 3.69 ∞
S LOBE 1.25 2.71 1.68 2.36 0.65 3.05 ∞
S SPUR 5.50 2.17 1.53 0.20 0.12 3.95 0.12
Lower and upper limits are quoted at∼90% confidence.
Table 4.3: Best-fitting parameters for the joint CPL and BPL fits to all six sub-regions of 3C449.
Best Fit Min Slope Max Slope
q f max χ 2 q − f max q + f max
joint CPL 2.68 1.67 33.5 2.55 1.30 2.81 2.00
Best Fit Min Slope Max Slope
q l f b q h χ 2 q − l
f b q − h
q + l
f b q + l
joint BPL 2.07 0.031 2.97 17.7 1.99 0.044 2.91 2.17 0.021 3.09
Lower and upper limits are quoted at∼90% confidence. The values of q and f max for the joint CPL fit and
q h , q l and f b for the joint BPL fit are the same for all sub-regions, while the normalizations are varied
to minimize the overallχ 2 . In the joint BPL fit, the maximum frequency is fixed at f max = 1.67 arcsec −1 .
the free parameters of the fit are the six normalizations, D 0 , one for each sub-region, the high
and low-frequency slopes, q h and q l , and the break and maximum spatial frequencies f b and
f max . I found best fitting parameters of q l = 2.07, q h =2.98, f b =0.031 arcsec −1 . As noted earlier, I
also fixed f max = 1.67 arcsec −1 to ensure consistency with the observed depolarizations at 1.25-
arcsec resolution. The corresponding structure functions are plotted in Fig. 4.7(g)–(l) and the
normalizations for the individual regions are given in Table 4.4. As for the CPL fits, the errors bars
are derived from the rms scatter of the structure functions of multiple convolved RM realizations.
It is evident from Fig. 4.7 that the structure functions corresponding to the BPL power
spectrum, which gives less power on large spatial scales, agree much better with the data. The
joint BPL fit has aχ 2 of 17.7, compared with 33.5 for the joint CPL fit (the former has only two
extra parameters), which confirms this result.
I have so far ignored the effects of any outer scale of the magnetic-field fluctuations. This
is justified because the structure functions for the spurs continue to rise at the largest observed
separations, indicating that the outer scale must be> ∼ 10 arcsec (≃30 kpc). The model structure
58

4.6. Three-dimensional analysis 59
Table 4.4: Normalizations and expected depolarization for the individual CPL, joint CPL and joint
BPL fit parameters at 1.25 arcsec.
Region Observed< k> CPL JOINT CPL JOINT BPL
C 0 f max < k> C 0 < k> D 0 < k>
[rad 2 m −4 ] [arcsec −1 ] [rad 2 m −4 ] [rad 2 m −4 ] [rad 2 m −4 )]
N LOBE 61±6 0.96 1.60 63±3 1.52 66±3 1.91 52±4
N JET 106±12 1.34 1.21 106±5 4.76 110±5 0.5 109±5
S JET 91±11 1.50 1.95 70±5 1.94 73±5 1.52 65±4
S LOBE 50±5 1.18 1.68 53±2 1.28 50±2 2.20 45±3
Col.1: region; Col. 2: observed Burn law< k>. Col. 3, 4 and 5: normalization constant C 0 , fitted maximum spatial
frequency f max for the best CPL power spectrum of each region and the predicted for
each power spectrum. Col. 6, 7 as Col. 3 and 5 but for the joint fit to each CPL power spectrum;
Col. 8 and 9 as Col. 3, and 5 but for the joint BPL power spectrum. For both the joint CPL and BPL fits,
the maximum frequency is fixed at f max = 1.67 arcsec −1 . In calculating each value of< k> only data with p>4σ p are included.
functions fit to the observations assume that the outer scale is infinite and the realizations are
generated on sufficiently large grids in Fourier space that the effects of the implicit outer scale
are negligible over the range of scales here sampled. I used structure-function data for the entire
source to determine an approximate value for the outer scale in Sec. 4.6.3.
I now adopt the BPL power spectrum with these parameters and investigate the spatial
variations of the RM fluctuation amplitude using three-dimensional simulations.
4.6 Three-dimensional analysis
4.6.1 Models
I used the software packageFARADAY (Murgia et al. 2004) to compare the observed RM with
simulated images derived from three-dimensional multi-scale magnetic-field models. Given a
field model and the density distribution of the thermal gas,FARADAY calculates an RM image by
integrating Eq. 3.2 numerically. As in Sec. 4.5, the fluctuations of RM were modeled on the
assumption that the magnetic field responsible for the foreground rotation is an isotropic, Gaussian
random variable and therefore characterized entirely by its power spectrum. Each point in a cube
in Fourier space was first assigned components of the magnetic vector potential. The amplitudes
were selected from a Rayleigh distribution of unit variance, and the phases were random in [0, 2π].
The amplitudes were then multiplied by the square root of the power spectrum of the vector
potential, which is simply related to that of the magnetic field. The corresponding components
of the magnetic field along the line-of-sight were then calculated and transformed to real space.
This procedure ensured that the magnetic field was divergence-free. The field components in real
59

60 4. The magneto-ionic medium around 3C 449
Table 4.5: Summary of magnetic field power spectrum and density scaling parameters.
BPL power spectrum
q l =2.07
low-frequency slope
q h =2.98
high-frequency slope
f b =0.031 arcsec −1
break frequency (Λ b = 1/ f b =11 kpc)
f max =1.67 arcsec −1
maximum frequency (Λ min = 1/ f max =0.2 kpc)
f min fitted minimum frequency (Λ max = 1/ f min )
Scaling of the magnetic field
B 0 fitted
Average magnetic field at group center
η fitted Magnetic field exponent of the radial profile:〈B〉(r)= B 1 1860 [ n e (r)
n 0
] η
space were then multiplied by the model density distribution and integrated along the line of sight
to give a synthetic RM image at the full resolution of the simulation, which was then convolved to
the observing resolution.
For 3C 449, I assumed that the source is in a plane perpendicular to the line- of-sight, which
passes through the group center and simulated the field and density structure using a 2048 3 cube
with a real-space pixel size of 0.1 kpc.
I used the best-fitting BPL power spectrum found in
Sec. 4.5.2, but with a spatially-variable normalization, as described below (Sec. 4.6.2), and a lowfrequency
cut-off f min , corresponding to a maximum scale of the magnetic field fluctuations, 1 Λ max
(= f −1
min ). The power spectrum of Eq. 4.4 was then set to 0 for f

4.6. Three-dimensional analysis 61
number of simulations for each combination of field strength and radial profile and to compare
the predicted and observed values ofσ RM evaluated over the boxes used in Sec. 4.3.1 (Fig. 4.2). I
usedχ 2 summed over the boxes as a measure of the goodness of fit. This procedure is independent
of the precise value of the outer scale, provided that it is much larger than the averaging boxes. I
express the results in terms ofχ 2 red , which is the value ofχ2 divided by the number of degrees of
freedom.
I initially tried a radial field-strength variation of the form
〈B 2 (r)〉 1/2 = B 0
[
ne (r)
n 0
] η
(4.5)
as used by Guidetti et al. (2008) and Laing et al. (2008). Here, B 0 is the rms magnetic field strength
at the group center and n e (r) is the thermal electron gas density, assumed to follow theβ-model
profile derived by Croston et al. (2008) (Sec. 4.1). As mentioned in Sec. 3.5, the field scaling
of Eq. 4.5 is consistent with other observations, analytical models and numerical simulations. In
particular,η=2/3 corresponds to flux-freezing andη=1/2 to equipartition between thermal and
magnetic energy.
I produced simulated RM images for each combination of B 0 andηin the ranges 0.5 – 10µG
in steps of 0.1µG and 0 – 2 in steps of 0.01, respectively. I then derived the syntheticσ RM profiles
and, by comparing them with the observed one, calculated the unweightedχ 2 . I repeated this
procedure 35 times at each angular resolution, noting the (B 0 ,η) pair that gave the lowestχ 2 in
each case. These values are plotted in Fig. 4.8. As in earlier work (Murgia et al. 2004; Guidetti
et al. 2008; Laing et al. 2008), I found a degeneracy between the values of B 0 andηin the sense
that the fitted values are positively correlated, but there are clear minima inχ 2 at both resolutions.
I therefore adopted the mean values of B 0 andη, weighted by 1/χ 2 , as the best overall estimates.
These are also plotted in Fig. 4.8 as blue crosses. Although the central magnetic field strengths
derived for the two RM images are consistent at the 1σ level (B 0 =2.8±0.5µG and B 0 = 4.1±1.2µG
at 5.5 and 1.25-arcsec resolution, respectively), the values ofηare not. The best-fitting values are
η=0.0±0.1 at 5.5 arcsec FWHM andη=0.8±0.4 at 1.25 arcsec FWHM.
I next produced 35 RM simulations at each angular resolution by fixing B 0 andηat their
best values for that resolution. The weightedχ 2 ’s for theσ RM profiles were calculated evaluating
the errors for each box by summing in quadrature the rms due to sampling (determined from the
dispersion in the realizations) and the fitting-error of the observations. These values are listed in
Table 4.6. The observed and best-fitting model profiles at both angular resolutions are shown in
Fig. 4.9.
A model withη ≃ 0 at all radii is a priori unlikely: previous work has found values of
61

62 4. The magneto-ionic medium around 3C 449
0.5< ∼ η< ∼ 1 in other sources (Dolag 2006; Guidetti et al. 2008; Laing et al. 2008). The most
likely explanation for the low value ofηinferred from the low-resolution image is that the electron
density distribution is not well represented by aβ-model (which describes a spherical and smooth
distribution) at large radii. In support of this idea, Fig. 4.1 shows that the morphology of the X-ray
emission is not spherical at large radii, but quite irregular. Croston et al. (2003, 2008) pointed out
that the quality of the fit of a singleβ-model to the X-ray surface brightness profile was poor in
the outer regions, suggesting small-scale deviations in the gas distribution. The singleβ-model
gave a better fit to the inner region of the X-ray surface brightness profile, where the polarized
emission of 3C 449 can be observed at 1.25-arcsec resolution. The aim is to fit theσ RM profiles
at both resolutions with the same distribution of n e (r)〈B(r) 2 〉 1/2 . In the rest of this subsection I
assume that the density profile n e (r) is still represented by the singleβ-model, even though I have
argued that it might not be appropriate in the outer regions of the hot gas distribution. Although
the resulting estimates of field strength at large radii may be unreliable, the fit is still necessary
for the calculation of the outer scale described in Sec. 4.6.3, which depends only on the combined
spatial variation of density and field strength.
The best-fitting model at 1.25-arcsec resolution, which is characterized by a more physically
reliableη, gives a very bad fit to the low resolution profile at almost all distances from the core
(Fig. 4.9a). Conversely, the model determined at 5.5-arcsec resolution gives a very poor fit to the
sharp peak inσ RM observed within 20 kpc of the nucleus at 1.25-arcsec resolution, where the radio
and X-ray data give the strongest constraints (Fig. 4.9b). I also verified that no single intermediate
value ofηgives an adequate fit to theσ RM profiles at all distances from the nucleus.
A better description of the observedσ RM profile is provided by the empirical function
〈B 2 (r)〉 1/2 = B 0
[
ne (r)
n 0
] ηint
r≤r m
= B 0
[
ne (r)
n 0
] ηout
r>r m ,
(4.6)
whereη int ,η out are the inner and outer scaling index of the magnetic field and r m is the break
radius.
I fixedη int =1.0 andη out =0.0, consistent with the initial results, in order to reproduce both the
inner sharp peak and the outer flat decline of theσ RM observed at the two resolutions, keeping
r m as a free parameter. I made three sets of three-dimensional simulations for values of the outer
scaleΛ max = 205, 65 and 20 kpc. Anticipating the result of Sec. 4.6.3, I plotted the results only for
Λ max = 65 kpc, but the derivedσ RM profiles are in any case almost independent of the value of the
outer scale in this range. The new simulations were made only at a resolution of 5.5 arcsec, since
62

4.6. Three-dimensional analysis 63
(a)
(b)
Figure 4.8: (a) and (b): Distributions of the best-fitting values of B 0 andηfrom 35 sets of
simulations, each covering ranges of 0.5 – 10µG in B 0 and 0 – 2 inη, at 5.5 and 1.25 arcsec
respectively. The sizes of the circles are proportional toχ 2 for the fit and the blue crosses represent
the means of the distributions weighted by 1/χ 2 . The plot shows the expected degeneracy between
B 0 andη.
(a)
(b)
Figure 4.9: (a): Observed and synthetic radial profiles for rms Faradayσ RM at 5.5 arcsec as
functions of the projected distance from the radio source center. The outer scale isΛ max =205 kpc.
The black points represent the data with vertical bars corresponding to the rms error of the RM fit.
The magenta crosses represent the mean values from 35 simulated profiles and the vertical bars
are the rms scatter in these profiles due to sampling. (b) as (a), but at 1.25 arcsec.
the larger field of view at this resolution is essential to define the change in slope of the profile.
In order to determine the best-fitting break radius, r m in Eq. 4.6, I produced 35 sets of synthetic
RM images for a grid of values of B 0 and r m for each outer scale, noting the pair of values which
gave the minimum unweightedχ 2 for theσ RM profile for each set of simulations. These values
are plotted in Fig. 4.10, which shows that there is a degeneracy between the break radius r m and
B 0 . As with the similar degeneracy between B 0 andηnoted earlier, there is a clear minimum in
χ 2 , and I therefore adopted the mean values of B 0 and r m weighted by 1/χ 2 as the best estimates
of the magnetic-field parameters.
63

64 4. The magneto-ionic medium around 3C 449
Table 4.6: Results from the three-dimensional fits at both the angular resolutions of 1.25 and
5.5 arcsec.
FWHM Λ max singleη brokenη
(arcsec) (kpc) B 0 (µG) η χ 2 red
B 0 (µG) r m (kpc) χ 2 red
1.25 205 4.1±1.1 0.8±0.4 0.48 - - -
5.50 205 2.8±0.5 0.1±0.1 2.2 3.5±0.7 17±9 1.9
5.50 65 - - - 3.5±1.2 16±11 1.8
5.50 20 - - - 3.5±0.8 11±8 2.1
I then made 35 simulations with the best-fitting values of B 0 andηfor each outer scale
and evaluated the weightedχ 2 ’s for the resultingσ RM profiles. All three values ofΛ max under
investigation give reasonable fits to the observedσ RM profile along the whole radio source. The fit
forΛ max = 65 kpc is marginally better than for the other two values (χ 2 red
= 1.8), consistent with
the results of Sec. 4.6.3 below. In this case the central magnetic field strength is 3.5±1.2µG and
the break radius is 16±11 kpc. For the power spectrum withΛ max = 65 kpc and these best-fitting
parameters, I also produced three-dimensional simulations at a resolution of 1.25 arcsec. Even
though the fitting procedure is based only on the low-resolution data, this model also reproduces
the 1.25-arcsec profile very well (χ 2 red =0.7). Combining the values ofχ2 for the two resolutions,
using the 1.25-arcsec profile close to the core and the 5.5-profile at larger distances, I findχ 2 red =1.8.
Figure 4.11 shows a comparison of the observed radial profiles for rms Faradayσ RM and
〈RM〉 with the synthetic ones derived for this model. The syntheticσ RM profile plotted in Fig. 4.11
is the mean over 35 simulations and may be compared directly with the observations. In contrast,
the〈RM〉 profile is derived from a single example realization. It is important to emphasize that the
latter is one example of a random process, and is not expected to fit the observations; rather, I aim
to compare the fluctuation amplitude as a function of position.
The values of B 0 andχ 2 red for all of the three-dimensional simulations, together withηand r m
for the single and double power-law profiles, respectively, are summarized in Table 4.6.
4.6.3 The outer scale of the magnetic-field fluctuations
The theoretical RM structure function for uniform field strength, density and path length and a
power spectrum with a low-frequency cut-off should asymptotically approach a constant value
(2σ 2 RM for a large enough averaging region) at separations > ∼ Λ max . The observed RM structure
function of the whole source is heavily modified from this theoretical one by the scaling of the
electron gas density and magnetic field at large separations, which acts to suppress power on large
spatial scales. In Sec. 4.5 I therefore limited the study of the structure function to sub-regions of
3C 449 where uniformity of field strength, density and path length (and therefore of the power-
64

4.6. Three-dimensional analysis 65
Figure 4.10: Distribution of the bestfitting
values of B 0 and r m from 35 sets
of simulations, each covering ranges of
0.5 – 10µG in B 0 and 0 – 50 kpc in r m .
The sizes of the circles are proportional
toχ 2 and the blue cross represents the
means of the distribution weighted by
1/χ 2 . The plot shows the degeneracy
between B 0 and r m described in the text.
spectrum amplitude) is a reasonable assumption, inevitably limiting our ability to constrain the
power spectrum on the largest scales.
Now that I have an adequate model for the variation of n e (r)〈B 2 (r)〉 1/2 with radius (Eq. 4.6), I
can correct for it to derive what I call the pseudo-structure function – that is the structure function
for a power-spectrum amplitude, which is constant over the source. This can be compared directly
with the structure functions derived from the Hankel transform of the power spectrum. To evaluate
the pseudo-structure function, I divided the observed 5.5-arcsec RM image by the function
[∫ L
1/2
n e (r) 2 B(r) dl] 2 (4.7)
0
(Enßlin & Vogt 2003), where the radial variations of n e and B are those of the best-fitting model
(Sec. 4.6) and the upper integration limit L has a length of 10 times the core radius. The integral
was normalized to unity at the position of the radio core: this is equivalent to fixing the field and
gas density at their maximum values and holding them constant everywhere in the group. The
normalization of the pseudo-structure function should then be quite close to that of the two central
jet regions.
The pseudo-structure function is shown in Fig. 4.12(a) together with the predictions for the
BPL power spectra withΛ max =205, 65 and 20 kpc. As expected, the normalization of the pseudostructure
function is consistent with that of the jets (Fig. 4.7a and b).
The comparison between the synthetic and observed pseudo-structure functions indicates
firstly that they agree very well at small separations, independent of the value ofΛ max . This
confirms that the best BPL power spectrum found from a combined fit to all six sub-regions
is a very good fit over the entire source. Secondly, despite the poor sampling on very large
scales, the asymptotic values of the predicted structure functions for the three values ofΛ max
are sufficiently different from each other that it is possible to determine an approximate outer
65

66 4. The magneto-ionic medium around 3C 449
(a)
(c)
(b)
(d)
Figure 4.11: Comparison between observed and synthetic profiles of rms and mean Faraday
rotation at resolutions of 5.5 arcsec (a and b) and 1.25 arcsec (c and d). The synthetic profiles
are derived from the best-fitting model withΛ max = 65 kpc. The black points represent the data
with vertical bars corresponding to the rms fitting error. (a) and (c): Profiles ofσ RM . The magenta
crosses represent the mean values from 35 simulations at 5.5-arcsec resolution, and the vertical
bars are the rms scatter in these profiles due to sampling. (b) and (d): Profiles of〈RM〉 derived
from single example realizations at 5.5 and 1.25-arcsec resolutions.
scale. The model withΛ max =65 kpc gives the best representation of the data. The fit is within
the estimated errors except for a marginal discrepancy at the very largest (and therefore poorly
sampled) separations. That withΛ max =20 kpc is inconsistent with the observed pseudo-structure
function for any separation> ∼ 20 arcsec (≃6 kpc) where the sampling is still very good, and is firmly
excluded. The model withΛ max =205 kpc has slightly, but significantly too much power on large
scales.
I emphasize that such estimate of the outer scale of the RM fluctuations is essentially
independent of the functional form assumed for the variation of field strength with radius in the
central region, which affects the structure function only for small separations. The results are
almost identical if I fit the field-strength variation with either the profile of Eq. 4.5 (withη≈0) or
that of Eq. 4.6.
As for the structure functions of individual regions, the pseudo-structure function at large
66

4.7. Summary and comparison with other sources 67
separations is clearly affected by poor sampling: this increases the errors, but does not produce
any bias in the derived values. At large radii, however, the integral in Eq. 4.7 becomes small, so
the noise on the RM image is amplified. This is a potential source of error, and I therefore checked
the results using numerical simulations. I calculated the mean and rms structure functions for sets
of realizations of RM images generated with theFARADAY code for different values ofΛ max . These
structure functions are plotted in Figs. 4.12(b) and (c).
The main difference between the model structure functions derived from simulations and the
pseudo-structure functions described earlier is that the former show a steep decline in power on
large scales in place of a plateau. This occurs because the smooth fall-off in density and magnetic
field strength with distance from the nucleus suppresses the fluctuations in RM on large scales.
The results of the simulations confirm the analysis using the pseudo-structure function. The
mean model structure function withΛ max = 65 kpc again fits the data very well, except for a
marginal discrepancy at the largest scales. In view of the deviations from spherical symmetry on
large scales evident in the X-ray emission surrounding 3C 449 (Fig. 4.1), I do not regard this as a
significant effect.
In order to check that the best-fitting density and field model also reproduces the data at
small separations, I repeated the analysis at 1.25-arcsec resolution. The observed pseudo-structure
function is shown in Fig. 4.12(d), together with the the predictions for the BPL power spectrum
withΛ max = 205, 65 and 20 kpc. The observed structure function is compared with the mean from
35 simulations withΛ max = 65 kpc in Fig. 4.12(e). In both cases, the agreement forΛ max = 65 kpc
is excellent.
In Fig. 4.13, example realizations of this model with the best-fitting field variation are shown
for resolutions of 1.25 and 5.5 arcsec alongside the observed RM images.
4.7 Summary and comparison with other sources
4.7.1 Summary
In this work I have studied the structure of the magnetic field associated with the ionized medium
around the radio galaxy 3C 449. I have analysed images of linearly polarized emission with
resolutions of 1.25 arcsec and 5.5 arcsec FHWM at seven frequencies between 1.365 and 8.385
GHz, and produced images of degree of polarization and rotation measure. The RM images at
both the angular resolutions show patchy and random structures. In order to study the spatial
statistics of the magnetic field, I used a structure-function analysis and performed two- and threedimensional
RM simulations. I can summarize the results as follows.
67

68 4. The magneto-ionic medium around 3C 449
(a)
(b)
(c)
(d)
(e)
Figure 4.12: (a): Comparison of the observed pseudo-structure function of the 5.5-arcsec RM
image as described in the text (points) with the predictions of the BPL model (curves, derived
using a Hankel transform). The predicted curves are forΛ max =205 (blue dotted), 65 (continuous
magenta) and 20 kpc (green dashed). (b) and (c): structure functions of the observed (points)
and synthetic 5.5-arcsec RM images produced with theΛ max =205,20 kpc andΛ max =65 kpc,
respectively. (d) and (e) as (a) and (c) respectively, but at 1.25-arcsec resolution. The error bars
are the rms from 35 structure functions at a givenΛ max , in (a) and (d) are representative error bars
from the model withΛ max =65 kpc.
68

4.7. Summary and comparison with other sources 69
1. The absence of deviations fromλ 2 rotation over a wide range of polarization position angle
implies that a pure foreground Faraday screen with no mixing of radio-emitting and thermal
electrons is a good approximation for 3C 449 (Sec. 4.3).
2. The best estimate for the Galactic contribution to the RM of 3C 449 is a constant value of
−160.7 rad m −2 (Sec. 4.3.2).
3. The dependence of the degree of polarization on wavelength is well fitted by a Burn
law. This is also consistent with pure foreground rotation, with the residual depolarization
observed at the higher resolution being due to unresolved RM fluctuations across the beam
(Sec. 4.4). There is no evidence for a detailed correlation of radio-source structure with
either RM or depolarization.
4. There is no obvious anisotropy in the RM distribution, consistent with the assumption that
the magnetic field is an isotropic, Gaussian random variable.
5. The RM structure functions for six different regions of the source are consistent with the
hypothesis that only the amplitude of the RM power spectrum varies across the source.
6. A broken power-law spectrum of the form given in Eq. 4.4 with q l = 2.07, q h =2.98,
f b =0.031 arcsec −1 and f max = 1.67 arcsec −1 (corresponding to a spatial scaleΛ min =0.2 kpc)
is consistent with the observed structure functions and depolarizations for all six regions.
No single power law provides a good fit to all of the structure functions.
7. The high-frequency cut-off in the power spectrum is required to model the depolarization
data.
8. The profiles ofσ RM strongly suggest that most of the fluctuating component of RM is
associated with the intra-group gas, whose core radius is comparable with the characteristic
scale of the profile (Sec. 4.3.1). The symmetry of the profile is consistent with the idea that
the radio source axis is close to the plane of the sky.
9. I therefore simulated the RM distributions expected for an isotropic, random magnetic field
in the hot plasma surrounding 3C 449, assuming the density model derived by Croston et al.
(2008).
10. These three-dimensional simulations show that the dependence of magnetic field on density
is best modeled by a broken power-law function with B(r)∝n e (r) close to the nucleus and
B(r)≈ constant at larger distances.
11. With this density model, the best estimate of the central magnetic field strength is B 0 =
3.5±1.2µG.
69

70 4. The magneto-ionic medium around 3C 449
12. Assuming these variations of density and field strength with radius, a structure-function
analysis can be used to estimate the outer scaleΛ max of the magnetic-field fluctuations. I
find excellent agreement forΛ max ≈ 65 kpc ( f min = 0.0053 arcsec −1 ).
4.7.2 Comparison with other sources
These results are qualitatively similar to those of Laing et al. (2008) on 3C 31. The maximum RM
fluctuation amplitudes are similar in the two sources, as are their environments. For sphericallysymmetric
gas density models, the central magnetic fields are almost the same: B 0 ≈ 2.8µG for
3C 31 and 3.5µG for 3C 449. Both results are consistent with the idea that the RM fluctuation
amplitude in galaxy groups and clusters scales roughly linearly with density, ranging from a
few rad m −2 in the much sparsest environments (e.g. NGC 315; Laing et al. 2006a), through
intermediate values≈30 – 100 rad m −2 in rich groups such as 3C 31 and 3C 449 to∼ 10 4 rad m −2
in the centers of clusters with cool cores.
The RM distribution of 3C 31 is asymmetrical, the northern (approaching) side of the source
showing a much lower fluctuation amplitude, consistent with the inclination of≈ 50 ◦ estimated by
Laing & Bridle (2002a). Detailed modeling of the RM profile led Laing et al. (2008) to suggest that
there is a cavity in the X-ray gas, but this would have to be significantly larger than the observed
extent of the radio lobes and is, as yet, undetected in X-ray observations. A broken power-law
scaling of magnetic field with density, similar to that found for 3C 449 in the present work, would
also improve the fit to theσ RM profile for 3C 31; alternatively, the effects of cavities around the
inner lobes and spurs of 3C 449 might be significant. Deeper X-ray observations of both sources
are needed to resolve this issue. In neither case is the magnetic field dynamically important: for
3C 449 the ratio of the thermal and magnetic-field pressures is≈30 at the nucleus and≈400 at
the core radius of the group gas, r c = 19 kpc. The magnetic field is therefore not dynamically
important, as in 3C 31.
The magnetic-field power spectrum in both sources can be fit by a broken power-law form.
The low-frequency slopes are 2.1 and 2.3 for 3C 449 and 3C 31 respectively. In both cases, the
power spectrum steepens at higher spatial frequencies, but for 3C 31 a Kolmogorov index (11/3)
provides a good fit, whereas I find that the depolarization data for 3C 449 require a cut-off below
a scale of 0.2 kpc and a high-frequency slope of 3.0. The break scales are≈5 kpc for 3C 31
and≈11 kpc for 3C 449. It is important to note that the simple parametrized form of the power
spectrum is not unique, and that a smoothly curved function would fit the data equally well.
The gas-density structure on large scales in the 3C 31 group is uncertain, so Laing et al. (2008)
could only give a rough lower limit to the outer scale of magnetic-field fluctuations,Λ max
> ∼ 70 kpc.
For 3C 449, we findΛ max ≈ 65 kpc. The projected distance between 3C 449 and its nearest
70

4.7. Summary and comparison with other sources 71
neighbour is≈33 kpc (Birkinshaw et al. 1981), similar to the scale on which the jets first bend
through large angles (Fig. 4.1). As in 3C 31, it is plausible that the outer scale of magnetic-field
fluctuations is set by interactions with companion galaxies in the group.
71

72 4. The magneto-ionic medium around 3C 449
IMAGE: RM_1.25_2000_SUBIM.FITS
IMAGE: SUBIM_1.25.FITS
;200 ;150 ;100
RAD/M/M
;200 ;150 ;100
RAD/M/M
Obs
Syn
IMAGE: RM_5.5_2000_SUBIM.FITS
IMAGE: SUBIM_5.5.FITS
;200 ;150 ;100
RAD/M/M
;200 ;150 ;100
RAD/M/M
Obs
Syn
Figure 4.13: Comparison of observed and representative synthetic distribution of Faraday RM at
1.25 arcsec and 5.5 arcsec. The synthetic images have been produced for the best-fitting model
withΛ max =65 kpc. The colour scale is the same for all displays.
72

Chapter 5
Ordered magnetic fields around radio
galaxies: evidence for interaction with
the environment †
M
ost of the RM images of radio galaxies published so far show patchy structures with no
clear preferred direction, consistent with isotropic foreground fluctuations over a range
of linear scales ranging from tens of kpc to< ∼ 100 pc (e.g. 3C 449, Chapter 4; see also Govoni et
al. 2006; Guidetti et al. 2008; Laing et al. 2008; Kuchar & Enßlin 2009; Bonafede et al. 2010),
Numerical modelling has demonstrated that this type of complex RM structure can be accurately
reproduced if the magnetic field is randomly variable with fluctuations on a wide range of spatial
scales, and is spread throughout the whole group or cluster environment (Chapter 4; see also
Murgia et al. 2004; Govoni et al. 2006; Guidetti et al. 2008; Laing et al. 2008; Bonafede et al.
2010). These authors used forward modelling, together with estimators of the spatial statistics of
the RM distributions (structure and autocorrelation functions or a multi-scale statistic) to estimate
the field strength, its relation to the gas density and its power spectrum. The technique of Bayesian
maximum likelihood has also been used for this purpose (Enßlin & Vogt 2005; Kuchar & Enßlin
2009).
In order to derive the three-dimensional magnetic field power spectrum, all of these authors
had to assume statistical isotropy for the field, since only the component of the magnetic field
along the line-of-sight contributes to the observed RM. This assumption is consistent with the
absence of a preferred direction in most of the RM images.
In contrast, the present Chapter reports on anisotropic RM structures, observed in lobed radio
galaxies located in different environments, ranging from a small group to one of the richest clusters
† Guidetti et al. 2011, MNRAS, in press
73

74 5. Ordered magnetic fields around radio galaxies
of galaxies. The RM images of radio galaxies presented in this Chapter show clearly anisotropic
“banded” patterns over part or all of their areas. In some sources, these banded patterns coexist
with regions of isotropic random variations. The magnetic field responsible for these RM patterns
must, therefore, have a preferred direction.
One source whose RM structure is dominated by bands is already known: M 84 (Laing &
Bridle 1987). In addition, there is some evidence for RM bands in sources which also show strong
irregular fluctuations, such as Cygnus A (Carilli & Taylor 2002). It is possible, however, that some
of the claimed bands could be due to imperfect sampling of an isotropic RM distribution with
large-scale power, and I return to this question in Section 5.8.1.
In this Chapter I present new RM images of three sources which show spectacular banded
structures, together with improved data for M 84. The environments of all four sources are
well characterized by modern X-ray observations, and I give the first comprehensive description
of the banded RM phenomenon. I present an initial attempt to interpret the phenomenon as a
consequence of source-environment interactions and to understand the difference between it and
the more usual irregular RM structure.
The RM images reported are derived from new or previously unpublished archive Very Large
Array data for the nearby radio galaxies 0206+35, M 84 (Laing et al. in preparation), 3C 270
(Laing, Guidetti & Bridle in preparation) and 3C 353 (Swain, private communication; see Swain
1996).
5.1 The Sample
High quality radio and X-ray data are available for all of the sources. In this Section I summarize
those of their observational properties which are relevant to this RM study. A list of the sources and
their general parameters is given in Table 5.1, while Table 5.2 shows the X-ray parameters taken
from the literature and equipartition parameters derived from radio observations in this work.
The sources were observed with the VLA at several frequencies, in full polarization mode and
with multiple configurations so that the radio structure is well sampled. The VLA observations,
data reduction and detailed descriptions of the radio structures are given for 0206+35 and M 84
by Laing, Guidetti et al. (in preparation), for 3C 270 by Laing, Guidetti & Bridle (in preparation),
and for 3C 353 by Swain (1996). All of the radio maps show a core, two sided jets and a doublelobed
structure with sharp brightness gradients at the leading edges of both lobes. The synchrotron
minimum pressures are all significantly lower than the thermal pressures of the external medium
(Table 5.2).
All of the sources have been observed in the soft X-ray band by more than one satellite,
74

5.1. The Sample 75
35 49 00
ROSAT PSPC
12 38 00
Chandra
35 48 30
DECLINATION (J2000)
35 48 00
35 47 30
DECLINATION (J2000)
12 37 00
12 36 00
35 47 00
(a)
12 35 00
(b)
02 09 44
02 09 40 02 09 36
RIGHT ASCENSION (J2000)
02 09 32
12 27 45
12 27 40 12 27 35
RIGHT ASCENSION (J2000)
12 27 30
05 52 00
XMM−Newton
;00 56 00
XMM−Newton
DECLINATION (J2000)
05 50 00
05 48 00
DECLINATION (J2000)
;00 57 00
;00 58 00
;00 59 00
05 46 00
12 19 40
12 19 30 12 19 20
RIGHT ASCENSION (J2000)
12 19 10
(c)
;01 00 00
;01 01 00
17 20 40
17 20 30
RIGHT ASCENSION (J2000)
17 20 20
(d)
Figure 5.1: X-ray images overlaid with radio contours for all sources: (a) 0206+35: 1385.1 MHz
VLA radio map with 1.2 arcsec FWHM; the contours are spaced by factor of 2 between 0.06 and
15 mJy beam −1 . The ROSAT PSPC image (Worrall & Birkinshaw 2000), is smoothed with a
Gaussian ofσ=30 arcsec. (b) M 84: 1413.0 MHz VLA radio map with 4.5 arcsec FWHM; the
contours are spaced by factor of 2 between 1 and 128 mJy beam −1 . The Chandra (Finoguenov
& Jones 2001) image is a wavelet reconstruction on angular scales from 4 up to 32 arcsec. (c)
3C 270: 1365.0 MHz VLA radio map with 5.0 arcsec FWHM; the contours are spaced by factor of
2 between 0.45 and 58 mJy beam −1 . The XMM-Newton image (Croston, Hardcastle & Birkinshaw
2005) is smoothed with a Gaussian ofσ=26 arcsec. (d) 3C 353: 1385.0 MHz VLA radio map
with 1.3 arcsec FWHM; the contours are spaced by factor of 2 between 0.35 and 22 mJy beam −1 .
The XMM-Newton image (Goodger et al. 2008) is smoothed with a Gaussian ofσ=30 arcsec.
The Chandra image of 3C 270 is displayed in logarithmic scale.
75

76 5. Ordered magnetic fields around radio galaxies
Table 5.1: General optical and radio properties: Col. 1: source name; Cols. 2&3: position; Col.
4: redshift; Col. 5: conversion from angular to spatial scale with the adopted cosmology; Col. 6:
Fanaroff-Riley class; Col. 7: the largest angular size of the radio source; Col. 8: radio power at
1.4 GHz; Col. 9: angle to the line of sight of the jet axis; Col. 10: environment of the galaxy; Col.
11: reference.
source RA DEC z kpc/arcsec FR class LAS log P 1.4 θ env. ref.
[J2000] [J2000] [arcsec] [W Hz −1 ] [degree]
0206+35 02 09 38.6+35 47 50 0.0377 0.739 I 90 24.8 40 group 1
3C 353 17 20 29.1 -00 58 47 0.0304 0.601 II 186 26.3 90 poor cluster 2
3C 270 12 19 23.2+05 49 31 0.0075 0.151 I 580 24.4 90 group 3
M 84 12 25 03.7+12 53 13 0.0036 0.072 I 150 23.2 60 rich cluster 3
References for the environmental classification: (1) Miller et al. (2002); (2) de Vaucouleurs et al. (1991); (3) Trager et al. (2000).
Table 5.2: X-ray and radio equipartition parameters for all the sources. Col. 1: source name; Col.
2: X-ray energy band; Col. 3: average thermal temperature; Cols. 4,5,6 and 7,8,9 best-fitting core
radii, central densities andβparameters for the outer and innerβmodels, respectively; Col. 10:
average thermal pressure at the midpoint of the radio lobes; Cols. 11&12: minimum synchrotron
pressure and corresponding magnetic field; Col.13: references for the X-ray models.
source band kT r cxout n 0out β out r cxin n 0in β in P 0 P min B Pmin ref.
[keV] [keV] [kpc] [cm −3 ] [kpc] [cm −3 ] [dyne cm −2 ] [dyne cm −2 ] µG
0206+35 0.2-2.5 1.30 +1.30
−0.30 22.2 2.4×10−3 0.35 0.85 0.42 0.70 9.6×10 −12 4.31× 10 −13 5.70 1, 2
3C 353 ” 4.33 +0.25
−0.24
1.66× 10 −12 11.2 3
3C 270 0.3-7.0 1.45 +0.23
−0.01 36.8 7.7×10−3 0.30 1.1 0.34 0.64 5.75× 10 −12 1.64× 10 −13 3.71 4
M 84 0.6-7.0 0.60 +0.05
−0.05
5.28±0.08 0.42 1.40±0.03 1.70× 10 −11 1.07× 10 −12 9.00 5
References: (1) Worrall & Birkinshaw (2000); (2)Worrall, Birkinshaw & Hardcastle (2001); (3) Iwasawa et al. (2000); (4) Croston et al. (2008);
(5) Finoguenov & Jones (2001).
allowing the detection of multiple components on cluster/group and sub-galactic scales. The X-
ray morphologies are characterized by a compact source surrounded by extended emission with
low surface brightness. The former includes a non-thermal contribution, from the core and the
inner regions of the radio jets and, in the case of 0206+35 and 3C 270, a thermal component
which is well fitted by a small core radiusβmodel. The latter component is associated with the
diffuse intra-group or intra-cluster medium. Parameters for all of the thermal components, derived
from X-ray observations, are listed in Table 5.2. Because of the irregular morphology of the hot
gas surrounding 3C 353 and M 84, it has not been possible to fitβmodels to their X-ray radial
surface brightness profiles.
76

5.1. The Sample 77
5.1.1 0206+35
0206+35 is an extended Fanaroff-Riley Class I (FR I; Fanaroff & Riley 1974) radio source whose
optical counterpart, UGC 1651, is a D-galaxy, a member of a dumb-bell system at the centre of
a group of galaxies. At a resolution of 1.2 arcsec the radio emission shows a core, with smooth
two-sided jets aligned in the NW-SE direction and surrounded by a diffuse and symmetric halo.
Laing & Bridle (in preparation) have estimated that the jets are inclined by≈40 ◦ with respect to
the line of sight, with the main (approaching) jet in the NW direction.
0206+35 has been observed with both the ROSAT PSPC and HRI instruments (Worrall &
Birkinshaw 1994, 2000; Trussoni et al. 1997) and with Chandra (Worrall, Birkinshaw & Hardcastle
2001). The X-ray emission consists of a compact source surrounded by a galactic atmosphere
which merges into the much more extended intra-group gas. The radius of the extended halo
observed by the ROSAT PSPC is≈2.5 arcmin (Fig. 5.1a). The ROSAT and Chandra X-ray surface
brightness profiles are well fit by the combination ofβmodels with two different core radii and a
power-law component (Hardcastle, private communication; Table 5.2).
5.1.2 3C 270
3C 270 is a radio source classified as FR I in most of the literature, although in fact, the two lobes
have different FR classifications at low resolution (Laing, Guidetti & Bridle in preparation). The
optical counterpart is the giant elliptical galaxy NGC 4261, located at the centre of a nearby group.
The radio source has a symmetrical structure with a bright core and twin jets, extending E-W and
completely surrounded by lobes. The low jet/counter-jet ratio indicates that the jets are close to
the plane of the sky, with the Western side approaching (Laing, Guidetti & Bridle in preparation).
The XMM-Newton image (Fig. 5.1c) shows a disturbed distribution with regions of low
surface-brightness (cavities) at the positions of both radio lobes. A recent Chandra observation
(Worrall et al. 2010) shows “wedges” of low X-ray surface brightness surrounding the inner jets
(see also Croston, Hardcastle & Birkinshaw 2005, Finoguenov et al. 2006, Jetha et al. 2007,
Croston et al. 2008). The overall surface brightness profile is accurately reproduced by a point
source convolved with the Chandra point spread function plus a doubleβmodel (Croston et al.
2008, projb model). Croston et al. 2008 found no evidence for a temperature gradient in the hot
gas. The group is characterized by high temperature and low luminosity (Finoguenov et al. 2006),
which taken together provide a very high level of entropy. This might be a further sign of a large
degree of impact of the AGN on the environment.
77

78 5. Ordered magnetic fields around radio galaxies
5.1.3 3C 353
3C 353 is an extended FR II radio source identified with a D-galaxy embedded at the periphery
of a cluster of galaxies. The best estimate for the inclination of the jets is≈90 ◦ (Swain, Bridle &
Baum 1998). The eastern jet is slightly brighter and ends in a well-defined hot spot. The radio
lobes have markedly different morphologies: the eastern lobe is round with sharp edges, while the
western lobe is elongated with an irregular shape. The location of the source within the cluster is of
particular interest for this work and might account for the different shapes of the lobes. Fig. 5.1(d)
shows the XMM-Newton image overlaid on the radio contours. The image shows only the NW
part of the cluster, but it is clear that the radio source lies on the edge of the X-ray emitting
gas distribution. so that the round eastern lobe is encountering a higher external density and is
probably also behind a larger column of Faraday-rotating material (Iwasawa et al. 2000, Goodger
et al. 2008). In particular, the image published by Goodger et al. (2008) shows that the gas density
gradient persists on larger scales.
5.1.4 M 84
M 84 is a giant elliptical galaxy located in the Virgo Cluster at about 400 kpc from the core. Optical
emission-line imaging shows a disk of ionized gas around the nucleus, with a maximum detected
extent of 20×7 arcsec 2 (Hansen, Norgaard-Nielsen & Jorgensen 1985; Baum et al. 1988; Bower et
al. 1997, 2000). The radio emission of M 84 (3C 272.1) has an angular extension of about 3 arcmin
(≃ 11 kpc) and shows an unresolved core in the nucleus of the galaxy, two resolved jets and a pair
of wide lobes (Laing & Bridle 1987; Laing et al. in preparation). The inclination to the line-ofsight
of the inner jet axis is∼60 ◦ , with the northern jet approaching, but there is a noticeable
bend in the counter-jet very close to the nucleus, which complicates modelling (Laing & Bridle
in preparation). After this bend, the jets remain straight for≈40 arcsec, then both of them bend
eastwards by∼90 ◦ and fade into the radio emission of the lobes.
The morphology of the X-ray emission has a H-shape made up of shells of compressed gas
surrounding cavities coincident with both the radio lobes (Finoguenov et al. 2008; Finoguenov et
al. 2006; Finoguenov & Jones 2001). This shape, together with the fact that the initial bending of
the radio jets has the same direction and is quite symmetrical, suggests a combination of interaction
with the radio plasma and motion of the galaxy within the cluster (Finoguenov & Jones 2001).
The ratio between the X-ray surface brightness of the shells of the compressed gas and their
surroundings is≈3 and is almost constant around the source. The shells are regions of enhanced
pressure and density and low entropy: the amplitude of the density enhancements (a factor of≈3)
suggests that they are produced by weak shock waves (Mach numberM∼1.3) driven by the
78

5.2. Analysis of RM and depolarization images 79
expanding lobes (Finoguenov et al. 2006).
5.2 Analysis of RM and depolarization images
The observational analysis is based on the following procedure. I first produced RM and Burn
law k images at two different angular resolutions for each source and searched for regions with
high k or correlated RM and k values, which could indicate the presence of internal Faraday
rotation and/or strong RM gradients across the beam (Secs. 3.2 and 3.3). In regions with low k
where the variations of RM are plausibly isotropic and random, I then used the structure function
(defined in Eq. 3.13) to derive the power spectrum of the RM fluctuations. Finally, to investigate
the depolarization in the areas of isotropic RM, and hence the magnetic field power on small scales,
I made numerical simulations of the Burn law k using the model power spectrum with different
minimum scales and compared the results with the data.
I assumed RM power spectra of the form:
Ĉ( f ⊥ ) = C 0 f −q
⊥ f ⊥ ≤ f max
= 0 f ⊥ > f max (5.1)
where f ⊥ is a scalar spatial frequency and fit the observed structure function (including the effect
of the observing beam) using the Hankel-transform method described by Laing et al. (2008) to
derive the amplitude, C 0 and the slope, q. To constrain the RM structure on scales smaller than the
beamwidth, I estimated the minimum scale of the best fitted field power spectrum,Λ min = 1/ f max ,
which predicts a mean value of k consistent with the observed one.
In this work, I am primarily interested in estimating the RM power spectrum over limited
areas, and I made no attempt to determine the outer scale of fluctuations.
5.3 Rotation measure images
The RM images and associated rms errors were produced by weighted least-squares fitting to
the observed polarization anglesΨ(λ) as a function ofλ 2 (Eq. 3.1) at three or four frequencies
(Table 5.3, see also the Appendix) using theRM task in theAIPS package.
Each RM map was calculated only at pixels with rms polarization-angle uncertainties

80 5. Ordered magnetic fields around radio galaxies
IMAGE: 02_RM_L_C_BLANK.FITS
0206+35
IMAGE: BLANK.FITS
M 84
;140 ;120 ;100 ;80 ;60 ;40 ;20
RAD/M/M −2
rad m
;20 0 20
RAD/M/M rad m −2
36 02 40
10 kpc
1.2 arcsec 4.5 arcsec
1 kpc
36 02 20
12 54 00
DECLINATION (J2000)
36 02 00
36 01 40
DECLINATION (J2000)
12 53 00
36 01 20
12 52 00
(a)
02 12 42
02 12 40 02 12 38
RIGHT ASCENSION (J2000)
02 12 36
(b)
12 25 08 12 25 04
RIGHT ASCENSION (J2000)
12 25 00
IMAGE: 3C270.RM.5.0_BLANKED.FITS
IMAGE: 3C353_RM_1.3_BLANK_BLANK_2000.FITS
3C 270 3C 353
05 52 00
;20 0 20 40
RAD/M/M −2
rad m
10 kpc
;50 0 50 100 150
RAD/M/M −2
rad m
;00 57 00
5 arcsec 1.3 arcsec
10 kpc
DECLINATION (J2000)
05 51 00
05 50 00
05 49 00
DECLINATION (J2000)
;00 58 00
;00 59 00
05 48 00
(c)
12 19 40
12 19 30 12 19 20
RIGHT ASCENSION (J2000)
12 19 10
;01 00 00
(d)
17 20 35
17 20 30 17 20 25
RIGHT ASCENSION (J2000)
17 20 20
Figure 5.2: RM images for all sources: (a) 0206+35; (b) M 84; (c) 3C 270; (d) 3C 353. The
angular resolution and the linear scale of each map are shown in the individual panels.
80

5.3. Rotation measure images 81
IMAGE: K;BURN_4p_BLANK.FITS
0206+35
IMAGE: K;BURN_3p_4.5_BLANKED.FITS
M 84
DECLINATION (J2000)
35 48 40
35 48 20
35 48 00
35 47 40
0 100 200 300 400 500 600
0 100 200 300 400 500 600
rad^2/m^4
rad^2/m^4
2
m−4
rad
2
m−4
0.92 0.82
DECLINATION (J2000)
12 54 00
12 53 30
12 53 00
12 52 30
35 47 20
(a)
12 52 00
(b)
02 09 42
02 09 40 02 09 38 02 09 36
RIGHT ASCENSION (J2000)
12 25 09
12 25 06 12 25 03
RIGHT ASCENSION (J2000)
12 25 00
IMAGE: K;BURN_4pBLANKED_0_5arcsec.FITS
3C 270
IMAGE: K;BURN_4p_BLANK.FITS
3C 353
DECLINATION (J2000)
05 52 00
05 51 00
05 50 00
05 49 00
05 48 00
0 100 200 300 400 500 600
rad^2/m^4 2 −4
m
(c)
DECLINATION (J2000)
;00 57 00
;00 58 00
;00 59 00
0 100 200 300 400 500 600
rad^2/m^4
2
m−4
0.83 (d)
0.78
12 19 40
12 19 30 12 19 20
RIGHT ASCENSION (J2000)
12 19 10
;01 00 00
17 20 35
17 20 30 17 20 25
RIGHT ASCENSION (J2000)
17 20 20
Figure 5.3: Burn law k images for all sources at the same angular resolutions as for the RM images:
(a) 0206+35 at 1.2 arcsec FWHM; (b) M 84 at 4.5 arcsec FWHM; (c) 3C 270 at 5 arcsec FWHM;
(d) 3C 353 at 1.3 arcsec FWHM. The corresponding integrated depolarization (DP; Section 5.4) is
indicated on the top right angle of each panel. The colour scale is the same for all displays.
81

82 5. Ordered magnetic fields around radio galaxies
Table 5.3: Frequencies, bandwidths and angular resolutions used in the RM and Burn law k images
discussed in Sects. 5.3 and 5.4, respectively.
source ν ∆ν beam
[MHz] [MHz] [arcsec]
0206+35 1385.1 25 1.2
1464.9 25
4885.1 50
3C 353 1385.0 12.5 1.3
1665.0 12.5
4866.3 12.5
8439.9 12.5
3C 270 1365.0 25 1.65
1412.0 12.5
4860.1 100
1365.0 25 5.0
1412.0 12.5
1646.0 25
4860.1 100
M 84 1413.0 25 1.65
4885.1 50
1385.1 50 4.5
1413.0 25
1464.9 50
4885.1 50
shown by Laing & Bridle (1987), but is derived from four-frequency data and has a higher signalto-noise
ratio.
In the fainter regions of 0206+35 (for which only three frequencies are available and the
signal-to-noise ratio is relatively low), the RM task occasionally failed to determine the nπ
ambiguities in position angle correctly. In order to remove these anomalies, I first produced
a lower-resolution, but high signal-to-noise RM image by convolving the 1.2 arcsec RM map
to a beamwidth of 5 arcsec FWHM. From this map I derived the polarization-angle rotations at
each of the three frequencies and subtracted them from the observed 1.2 arcsec polarization angle
maps at the same frequency to derive the residuals at high resolution. Then, I fit the residuals
without allowing any nπ ambiguities and added the resulting RM’s to the values determined at
low resolution. This procedure allowed us to obtain an RM map of 0206+35 free of significant
deviations fromλ 2 rotation and fully consistent with the 1.2-arcsec measurements.
I have verified that the polarization angles accurately follow the relation∆Ψ∝λ 2 over the
full range of position angle essentially everywhere except for small areas around the opticallythick
cores: representative plots ofΨagainstλ 2 for 0206+35 are shown in Fig. 5.4. The lack of
deviations fromλ 2 rotation in all of the radio galaxies is fully consistent with the assumption that
the Faraday rotating medium is mostly external to the sources.
The RM maps are shown in Fig. 5.2. The typical rms error on the fit is≈2 rad m −2 . No
82

5.3. Rotation measure images 83
correction for the Galactic contribution has been applied.
All of the RM maps show two-dimensional patterns, RM bands, across the lobes with
characteristic widths ranging from 3 to 12 kpc. Multiple bands parallel to each other are observed
in the western lobe of 0206+35, the eastern lobe of 3C 353 and the southern lobe of M 84.
In all cases, the iso-RM contours are straight and perpendicular to the major axes of the lobes
to a very good approximation: the very straight and well-defined bands in the eastern lobes of
both 0206+35 (Fig. 5.2a) and 3C 353 (Fig. 5.2d) are particularly striking. The entire area of M 84
appears to be covered by a banded structure, while in the central parts of 0206+35 and 3C 270 and
the western lobe of 3C 353, regions of isotropic and random RM fluctuations are also present.
I also derived profiles of〈RM〉 along the radio axis of each source, averaging over boxes
a few beamwidths long (parallel to the axes), but extended perpendicular to them to cover the
entire width of the source. The boxes are all large enough to contain many independent points.
The profiles are shown in Fig. 5.5. For each radio galaxy, I also plot an estimate of the Galactic
contribution to the RM derived from a weighted mean of the integrated RM’s for non-cluster radio
sources within a surrounding area of 10 deg 2 (Simard-Normandin et al. 1981). In all cases, both
positive and negative fluctuations with respect to the Galactic value are present.
In 0206+35 (Fig. 5.2a), the largest-amplitude bands are in the outer parts of the lobes, with
a possible low-level band just to the NW of the core. The most prominent band (with the most
negative RM values) is in the eastern (receding) lobe, about 15 kpc from the core (Fig. 5.5a). Its
amplitude with respect to the Galactic value is about 40 rad m −2 . This band must be associated with
a strong ordered magnetic field component along the line of sight. If corrected for the Galactic
contribution, the two adjacent bands in the eastern lobe would have RM with opposite signs and
the field component along the line of sight must therefore reverse.
M 84 (Fig. 5.2b) displays an ordered RM pattern across the whole source, with two wide
bands of opposite sign having the highest absolute RM values. There is also an abrupt change of
sign across the radio core (see also Laing & Bridle 1987). The negative band in the northern lobe
(associated with the approaching jet) has a larger amplitude with respect to the Galactic value than
the corresponding (positive) feature in the southern lobe (Fig. 5.5c).
3C 270 (Fig. 5.2c) shows two large bands: one on the front end of the eastern lobe, the other
in the middle of the western lobe. The bands have opposite signs and contain the extreme positive
and negative values of the observed RM. The peak positive value is within the eastern band at the
extreme end of the lobe (Fig. 5.5e).
The RM structure of 3C 353 (Fig. 5.2d) is highly asymmetric. The eastern lobe shows a
strong pattern, made up of four bands, with very straight iso-RM contours which are almost
83

84 5. Ordered magnetic fields around radio galaxies
Table 5.4: Properties of the RM bands: Col. 1 source name; Cols. 2&3: overall〈RM〉 andσ RM ;
Col. 4: Galactic〈RM〉 ; Col. 5: 〈RM〉 for each band; Col. 6: distance of the band midpoint
from the radio core (positive distances are in the western direction for all sources, except for M 84,
where they are in the northern direction); Col. 7: width of the band; Col. 8: maximum band
amplitude.
source 〈RM〉 σ RM RM G band〈RM〉 d c width A
[rad m −2 ] [rad m −2 ] [kpc] [kpc] [rad m −2 ]
0206+35 −77 23 −72
−140 -15 10 40
−60 -27 4
34 22 6
51 8 4
3C 353 −56 24 −69
122 -12 5 50
102 -19 4
-40 -23 4
100 -26 4
3C 270 14 10 12
−8 20 12 10
32 37 11
M 84 −2 15 2
−27 1 3 10
22 -6 6
exactly perpendicular to the source axis. As in 0206+35, adjacent bands have RM with opposite
signs once corrected for the Galactic contribution (Fig. 5.5g). In contrast, the RM distribution
in the western lobe shows no sign of any banded structure, and is consistent with random
fluctuations superimposed on an almost linear profile. It seems very likely that the differences
in RM morphology and axial ratio are both related to the external density gradient (Fig. 5.1d).
In Table 5.4 the relevant geometrical features (size, distance from the radio core,〈RM〉 ) for
the RM bands are listed.
5.4 Depolarization
In this section, I use “depolarization” in its conventional sense to mean “decrease of degree of
polarization with increasing wavelength” and define DP= p 1.4 GHz /p 4.9 GHz . Using theFARADAY
code (Murgia et al. 2004), I produced images of Burn law k by weighted least-squares fitting to
ln p(λ) as a function ofλ 4 (Eq. 3.5). Only data with signal-to-noise ratio>4 in P at each frequency
were included in the fits. The Burn law k images were produced with the same angular resolutions
as the RM images. The 1.65 arcsec resolution Burn law k maps for M 84 and 3C 270 are consistent
with the low-resolution ones, but add no additional detail and are quite noisy. This could lead
to significantly biased estimates for the mean values of k over large areas (Laing et al. 2008).
84

5.4. Depolarization 85
Therefore, as for the RM maps, I used only the Burn law k images at low resolution for these two
sources.
The Burn law k maps are shown in Fig. 5.3. All of the sources show low average values of k
(i.e. slight depolarization), suggesting little RM power on small scales. With the possible exception
of the narrow filaments of high k in the eastern lobe of 3C 353 (which might result from partially
resolved RM gradients at the band edges), none of the images show any obvious structure related
to the RM bands. For each source, I have also compared the RM and Burn law k values derived by
averaging over many small boxes covering the emission, and I find no correlation between them.
I also derived profiles of k (Fig. 5.5b, d, f and h) with the same sets of boxes as for the RM
profiles in the same Figure. These confirm that the values of k measured in the centres of the RM
bands are always low, but that there is little evidence for any detailed correlation.
The signal-to-noise ratio for 0206+35 is relatively low compared with that of the other three
sources, particularly at 4.9 GHz (a small beam is necessary to resolve the bands), and this is
reflected in the high proportion of blanked pixels on the k image. The most obvious feature of
this image (Fig. 5.3a), an apparent difference in mean k between the high-brightness jets (less
depolarized) and the surrounding emission, is likely to be an artefact caused by the adopted
blanking strategy: points where the polarized signal is low at 4.9 GHz are blanked preferentially,
so the remainder show artificially high polarization at this frequency. For the same reason, the
apparent minimum in k at the centre of the deep, negative RM band (Fig. 5.5a and b) is probably
not significant. The averaged values of k for 0206+35 are already very low, however, and are
likely to be slightly overestimated, so residual RM fluctuations on scales below the 1.2-arcsec
beamwidth must be very small.
M 84 shows one localised area of very strong depolarization (k∼500 rad 2 m −4 , corresponding
to DP=0.38) at the base of the southern jet (Fig. 5.3b). There is no corresponding feature in
the RM image (Fig. 5.2b). The depolarization is likely to be associated with one of the shells
of compressed gas visible in the Chandra image (Fig. 5.1b), implying significant magnetization
with inhomogeneous field and/or density structure on scales much smaller than the beamwidth,
apparently independent of the larger-scale field responsible for the RM bands. This picture is
supported by the good spatial coincidence of the high k region with a shell of compressed gas,
as illustrated in the overlay of the 4.5 arcsec Burn law k image on the contours of the Chandra
data (Fig. 5.6(a)). Cooler gas associated with the emission-line disk might also be responsible, but
there is no evidence for spatial coincidence between enhanced depolarization and Hα emission
(Hansen, Norgaard-Nielsen & Jorgensen 1985). Despite the complex morphology of the X-ray
emission around M 84, its k profile is very symmetrical, with the highest values at the centre
(Fig. 5.5(d)).
85

86 5. Ordered magnetic fields around radio galaxies
3C 270 also shows areas of very strong depolarization (k∼550 rad 2 m −4 , corresponding to
DP=0.35) close to the core and surrounding the inner and northern parts of both the radio
lobes. As for M 84, the areas of high k are coincident with ridges in the X-ray emission which
form the boundaries of the cavity surrounding the lobes (Fig. 5.6(b)). The inner parts of this X-
ray structure are described in more detail by Worrall et al. (2010), whose recent high-resolution
Chandra image clearly reveals “wedges” of low brightness surrounding the radio jets. As in M 84,
the most likely explanation is that a shell of denser gas immediately surrounding the radio lobes
is magnetized, with significant fluctuations of field strength and density on scales smaller than
the 5-arcsec beam, uncorrelated with the RM bands. The k profile of 3C 270 (Fig. 5.5(f)) is very
symmetrical, suggesting that the magnetic-field and density distributions are also symmetrical and
consistent with an orientation close to the plane of the sky. The largest values of k are observed in
the centre, coincident with the features noted earlier and with the bulk of the X-ray emission (the
high k values in the two outermost bins have low signal-to-noise and are not significant).
In the Burn law k image of 3C 353, there is evidence for a straight and knotty region of high
depolarization≈20 kpc long and extending westwards from the core. This region does not appear
to be related either to the jets or to any other radio feature. As in M 84 and 3C 270, the RM appears
quite smooth over the area showing high depolarization, again suggesting that there are two scales
of structure, one much smaller than the beam, but producing zero mean RM and the other very
well resolved. In 3C 353, there is as yet no evidence for hot or cool ionized gas associated with
the enhanced depolarization (contamination from the very bright nuclear X-ray emission affects
an area of 1 arcmin radius around the core; Iwasawa et al. 2000, Goodger et al. 2008).
The k profile of 3C 353 (Fig. 5.5(h)) shows a marked asymmetry, with much higher values in
the East. This is in the same sense as the difference of RM fluctuation amplitudes (Fig. 5.5(g))
and is also consistent with the eastern lobe being embedded in higher-density gas. The relatively
high values of k within 20 kpc of the nucleus in the Western lobe are due primarily to the discrete
region identified earlier.
5.5 Rotation measure structure functions
I calculated RM structure function (Eq. 3.13) for discrete regions of the sources where the RM
fluctuations appear to be isotropic and random and for which we expect the spatial variations of
foreground thermal gas density, rms magnetic field strength and path length to be reasonably small.
These are: the inner 26 arcsec of the receding (Eastern) lobe of 0206+35, the inner 100 arcsec of
3C 270 and the inner 40 arcsec of the western lobe of 3C 353. The selected areas of 0206+35
and 3C 270 are both within the core radii of the larger-scale beta models that describe the group-
86

5.5. Rotation measure structure functions 87
Table 5.5: Power spectrum parameters for the individual sub-regions. Col. 1: source name; Col.
2: angular resolution; Col. 3: slope q: Col. 4: amplitude C 0 ; Col. 5: minimum scale; Col. 6:
amplitude of the large scale isotropic component; Col. 7: observed mean k; Col. 8: predicted
mean k. The power spectrum has not been computed for M 84 (see Section 5.5).
source FWHM q log C 0 Λ min A iso k obs k syn
[arcsec] [kpc] [rad m −2 ] [rad 2 m −4 ] [rad 2 m −4 ]
0206+35 1.2 2.1 0.77 2 10 37 40
3C 270 1.65 2.7 0.90 0.1 5 30 26
5.0 2.7 0.90 0.1 5 71 64
3C 353 1.3 3.1 0.99 0.1 10 38 33
M 84 1.65

88 5. Ordered magnetic fields around radio galaxies
0206+35
Figure 5.4: Plots of E-vector position angleΨagainstλ 2 at representative points of the 1.2 arcsec
RM map of 0206+35. Fits to the relationΨ(λ)=Ψ 0 + RMλ 2 are shown. The values of RM are
given in the individual panels.
In order to constrain RM structure on spatial scales below the beamwidth, I estimated the
depolarization as described in Section 5.2. The fitted k values are listed in Table 5.5. I stress that
these values refer only to areas with isotropic fluctuations, and cannot usefully be compared with
the integrated depolarizations quoted in in Fig. 5.2.
For M 84, using the Burn law k analysis and assuming that variation of Faraday rotation
across the 1.65-arcsec beam causes the residual depolarization, I find thatΛ min
< ∼ 0.1 kpc for any
reasonable RM power spectrum.
5.6 Rotation-measure bands from compression
It is clear from the fact that the observed RM bands are perpendicular to the lobe axes that they
must be associated with an interaction between an expanding radio source and the gas immediately
surrounding it. One inevitable mechanism is enhancement of field and density by the shock or
compression wave surrounding the source. 1
The implication of the presence of cavities in the
X-ray gas distribution coincident with the radio lobes is that the sources are interacting strongly
1 An alternative mechanism is the generation of non-linear surface waves (Bicknell, Cameron & Gingold 1990). It
is unlikely that this can produce large-scale bands, for the reasons given in Section 5.8.5.
88

5.6. Rotation-measure bands from compression 89
0206+35
M 84
3C 270 3C 353
Figure 5.5: Profiles of mean and rms RM and Burn law k along the radio axis. The profiles
have been derived by averaging over boxes perpendicular to the radio axis of length 5 kpc for all
sources except M 84, for which the box length is 2 kpc. The orange (dashed) and violet (longdashed)
lines represent the Galactic RM contribution and the〈RM〉 of the lobes, respectively. All
unblanked pixels are included.
89

90 5. Ordered magnetic fields around radio galaxies
IMAGE: K;BURN_3p_4.5_BLANKED.FITS
CNTRS: M84_CHANDRA_NOPTS_AS_1.3_4.5.FITS.3
0 100 200 300 400 500 600
IMAGE: 3C270_KBURN_5.0.FITS
CNTRS: XMM_12_ARCSEC_AS_1.3_5.0.FITS
rad^2/m^4
m
0 100 200 300 400 500 600
rad^2/m^4
rad m
12 54 00
05 52 00
DECLINATION (J2000)
12 53 00
DECLINATION (J2000)
05 50 00
05 48 00
12 52 00
(a)
12 25 09
12 25 06 12 25 03
RIGHT ASCENSION (J2000)
12 25 00
(b)
12 19 40
12 19 30 12 19 20
RIGHT ASCENSION (J2000)
12 19 10
Figure 5.6: Burn law k images of M 84 (a) and 3C 270 (b) overlaid on X-ray contours derived
from Chandra and XMM-Newton data, respectively.
with the thermal gas, displacing rather than mixing with it (see McNamara & Nulsen 2007 for
a review). For the sources here presented, the X-ray observations of M 84 (Finoguenov et al.
2008, Fig.5.1b) and 3C 270 (Croston et al. 2008, Fig.5.1c) show cavities and arcs of enhanced
brightness, corresponding to shells of compressed gas bounded by weak shocks. The strength of
any pre-existing field in the IGM, which will be frozen into the gas, will also be enhanced in the
shells. Therefore a significant enhancement in RM is expected. A more extreme example of this
effect will occur if the expansion of the radio source is highly supersonic, in which case there will
be a strong bow-shock ahead of the lobe, behind which both the density and the field become much
higher. Regardless of the strength of the shock, the field is modified so that only the component in
the plane of the shock is amplified and the post-shock field tends become ordered parallel to the
shock surface.
The evidence so far suggests that shocks around radio sources of both FR classes are generally
weak (e.g. Forman et al. 2005, Wilson, Smith & Young 2006, Nulsen et al. 2005a). There are
only two examples in which highly supersonic expansion has been inferred: the southern lobe of
Centaurus A (M≈8; Kraft et al. 2003) and NGC 3801 (M≈4; Croston et al. 2007). There
is no evidence that the sources here described are significantly over-pressured compared with
the surrounding IGM (indeed, the synchrotron minimum pressure is systematically lower than the
thermal pressure of the IGM; Table 5.2). The sideways expansion of the lobes is therefore unlikely
to be highly supersonic. The shock Mach number estimated for all the sources from ram pressure
balance in the forward direction is also≈1.3. This estimate is consistent with that for M 84 made
by Finoguenov et al. (2006) and also with the lack of detection of strong shocks in the X-ray data
90

5.6. Rotation-measure bands from compression 91
(a)
(b)
(c)
(d)
Figure 5.7: Plots of the RM structure functions for the isotropic sub-regions of 0206+35,3C 353
and 3C 270 described in the text. The horizontal bars represent the bin width and the crosses
the centroids for data included in the bins. The red lines are the predictions for power law
power spectra, including the effects of the convolving beam. The vertical error bars are the
rms variations for the structure functions derived for multiple realizations of the data. The fitted
structure functions for 3C 270 are derived for the same power spectrum parameters.
for the other sources.
In this section, I investigate how the RM could be affected by compression. I consider a
deliberately oversimplified picture in which the radio source expands into an IGM with an initially
uniform magnetic field, B. This is the most favourable situation for the generation of large-scale,
anisotropic RM structures: in reality, the pre-existing field is likely to be highly disordered, or
even isotropic, because of turbulence in the thermal gas. I stress that I have not tried to generate a
self-consistent model for the magnetic field and thermal density, but rather to illustrate the generic
effects of compression on the RM structure.
In this model the radio lobe is an ellipsoid with its major axis along the jet and is surrounded
by a spherical shell of compressed material. This shell is centred at the mid-point of the lobe
(Fig. 5.8) and has a stand-off distance equal to 1/3 of the lobe semi-major axis at the leading
91

92 5. Ordered magnetic fields around radio galaxies
Figure 5.8: Amplification and sweeping up of the magnetic field lines inside the compression
region defined by the projected circular green and dashed line. The driving expansion is along the
Z-axis and the z-axis represents a generic line-of-sight.
edge (the radius of the spherical compression is therefore equal to 4/3 of the lobe semi-major
axis). In the compressed region, the thermal density and the magnetic field component in the
plane of the spherical compression are amplified by the same factor, because of flux-freezing. I
use a coordinate system xyz centred at the lobe mid-point, with the z-axis along the line of sight,
so x and y are in the plane of the sky. The radial unit vector is ˆr=(ˆx, ŷ, ẑ). B=(B x , B y , B z ) and
B ′ =(B ′ x, B ′ y, B ′ z) are respectively the pre- and post-shock magnetic-field vectors. Then, I consider
a coordinate system XYZ still centred at the lobe mid-point, but rotated with respect to the xyz
system by the angleθabout the y (Y) axis, so that Z is aligned with the major axis of the lobe.
With this choice,θis the inclination of the source with respect to the line-of-sight (Fig. 5.8).
After a spherical compression, the thermal density and field satisfy the equations:
n ′ e
= γn e
B ′ ⊥ = B ⊥ = (B·r)ˆr
B ′ ‖
= γB ‖ =γ[B−(B·r)ˆr] (5.2)
where n e , B ⊥ and B ‖ represent the initial thermal density and the components of the field
perpendicular and parallel to the compression surface. The same symbols with primes stand for
post-shock quantities andγis the compression factor. The total compressed magnetic field is:
B ′ = B ′ ‖ + B′ ⊥ =γB+(1−γ)(B·r)ˆr (5.3)
92

5.6. Rotation-measure bands from compression 93
The field strength after compression depends on the the angle between the compression surface and
the initial field. Maximum amplification occurs for a field which is parallel to the surface, whereas
a perpendicular field remains unchanged. The post-shock field component along the line-of-sight
becomes:
B ′ z = B′· ẑ=γB·ẑ+(1−γ)(B·r)(r·ẑ) (5.4)
I assumed that the compression factor,γ=γ(Z), is a function of distance Z along the source
axis from the centre of the radio lobe, decreasing monotonically from a maximum valueγ max at
the leading edge to a constant value from the centre of the lobe as far as the core. I investigated
values ofγ max in the range 1.5 – 4 (γ=4 corresponds to the asymptotic value for a strong shock).
Given that there is no evidence for strong shocks in the X-ray data for any of the sources under
investigation, I have typically assumed that the compression factor isγ max = 3 at the front end of
the lobe, decreasing to 1.2 at the lobe mid-point and thereafter remaining constant as far as the
core. A maximum compression factor of 3 is consistent with the transonic Mach numbersM≃1.3
estimated from ram-pressure balance for all of the sources and this choice is also motivated by the
X-ray data of M 84, from which there is evidence for a compression ratio≈ 3 between the shells
and their surroundings (Section 5.1.4).
I produced synthetic RM images for different combinations of source inclination and direction
of the pre-existing uniform field, by integrating the expression
RM syn =
∫ R0
lobe
n ′ eB ′ zdz. (5.5)
numerically. I assumed a constant value of n e = 2.4×10 −3 cm −3 for the density of the preshock
material (the central value for the group gas associated with 0206+35; Table 5.2), a lobe
semi-major axis of 21 kpc (also appropriate for 0206+35) and an initial field strength of 1µG.
The integration limits were defined by the surface of the radio lobe and the compression surface.
This is equivalent to assuming that there is no thermal gas within the radio lobe, consistent with
the picture suggested by the inference of foreground Faraday rotation and the existence of X-ray
cavities and that Faraday rotation from uncompressed gas is negligible.
As an example, Fig. 5.9 shows the effects of compression on the RM for the receding lobe
of a source inclined by 40 ◦ to the line of sight. The initial field is pointing towards us with an
inclination of 60 ◦ with respect to the line-of-sight; its projection on the plane of the sky makes
an angle of 30 ◦ with the x-axis. Fig. 5.9(a) displays the RM produced without compression
(γ(Z)=1 everywhere): the RM structures are due only to differences in path length across the
lobe. Fig. 5.9(b) shows the consequence of adding a modest compression ofγ max = 1.5: structures
similar to bands are generated at the front end of the lobe and the range of the RM values is
93

94 5. Ordered magnetic fields around radio galaxies
IMAGE: ICM.FITS
IMAGE: ICM_WEAKSHOCK.FITS
IMAGE: ICM_STRONGSHOCK.FITS
34 36 38 40 42 44
RAD/M/M
rad m −2
40 60 80 100 120 140 160
RAD/M/M−2
rad m
100 200 300 400 500
RAD/M/M−2
rad m
X X X
(a) (b) (c)
Figure 5.9: Panel (a): synthetic RM for a receding lobe inclined by 40 ◦ to the line-of-sight and
embedded in a uniform field inclined by 60 ◦ to the line-of-sight and by and 30 ◦ to the lobe axis on
the plane of the sky. Panels (b) and (c): as (a) but with a weak compression (γ max = 1.5) and a
strong compression (γ max = 4) of the density and field. The crosses and the arrows indicate the
position of the core and the lobe advance direction, respectively.
increased. Fig. 5.9(c) illustrates the RM produced in case of the strongest possible compression,
γ max = 4: the RM structure is essentially the same as in Fig. 5.9(b), with a much larger range.
This very simple example shows that RM bands with amplitudes consistent with those
observed can plausibly be produced even by weak shocks in the IGM, but the iso-RM contours
are neither straight, nor orthogonal to the lobe axis and there are no reversals. These constraints
require specific initial conditions, as illustrated in Fig. 5.10, where I show the RM for a lobe in the
plane of the sky. I considered three initial field configurations: along the line-of-sight (Fig. 5.10a),
in the plane of the sky and parallel to the lobe axis (Fig. 5.10b) and in the plane of the sky, but
inclined by 45 ◦ to the lobe axis (Fig. 5.10c). The case closest to reproducing the observations is
that displayed in Fig. 5.10(b), in which reversals and well defined and straight bands perpendicular
to the jet axis are produced for both of the lobes. In Fig. 5.10(a), the structures are curved, while
in Fig. 5.10(c) the bands are perpendicular to the initial field direction, and therefore inclined with
respect to the lobe axis.
For a source inclined by 40 ◦ to the line of sight, I found structures similar to the observed bands
only with an initial field in the plane of the sky and parallel to the axis in projection (Figs 5.11a
and b; note that the synthetic RM images in this example have been made for each lobe separately,
neglecting superposition).
I can summarize the results of the spherical pure compression model as follows.
1. An initial field with a component along the line-of-sight does not generate straight bands.
2. The bands are orthogonal to the direction of the initial field projected on the plane of the
sky, so bands perpendicular to the lobe axis are only obtained with an initial field aligned
with the radio jet in projection.
94

5.7. Rotation-measure bands from a draped magnetic field 95
IMAGE: BLANK.FITS
IMAGE: BLANK.FITS
IMAGE: BLANK.FITS
100 200 300 400
0 20 40 60 80 100 120
0 20 40 60 80 100
RAD/M/M−2
rad m
RAD/M/M−2
rad m
RAD/M/M−2
rad m
x
x
x
(a)
(b)
(c)
Figure 5.10: Synthetic RM images for a lobe in the plane of the sky with an ambient field which
is uniform before compression. In panels (a) and (b), the field is along the axis of the lobe in
projection and inclined to the line-of-sight by (a) 45 ◦ and (b) 90 ◦ . In panel (c), the field is in the
plane of the sky and misaligned by 45 ◦ with respect to the lobe axis. The crosses and the arrows
represent the radio core position and the lobe advance direction, respectively.
IMAGE: BLANK.FITS
0 20 40 60 80
RAD/M/M
rad m −2
IMAGE: BLANK.FITS
;140 ;120 ;100 ;80 ;60 ;40 ;20 0
RAD/M/M
rad m −2
x
x
(a)
(b)
Figure 5.11: Synthetic RM images of the receding (a) and approaching lobe (b) of a source
inclined by 40 ◦ with respect to the line-of-sight, in the case of a uniform ambient field. The
field is aligned with the radio jets in projection and inclined with respect to the line-of-sight by
90 ◦ . The crosses and the arrows represent the radio core position and the lobe advance direction,
respectively.
3. The path length (determined by the precise shape of the radio lobes) has a second-order
effect on the RM distribution (compare Figs. 5.9a and b).
Thus, a simple compression model can generate bands with amplitudes similar to those
observed but reproducing their geometry requires implausibly special initial conditions, as I
discuss in the next section.
5.7 Rotation-measure bands from a draped magnetic field
5.7.1 General considerations
That the pre-existing field is uniform, close to the plane of the sky and aligned with the source axis
in projection is implausible for obvious reasons:
95

96 5. Ordered magnetic fields around radio galaxies
1. the pre-existing field cannot know anything about either the radio-source geometry or our
line-of-sight and
2. observations of Faraday rotation in other sources and the theoretical inference of turbulence
in the IGM both require disordered initial fields.
This suggests that the magnetic field must be aligned by rather than with the expansion of the radio
source. Indeed, the field configurations which generate straight bands look qualitatively like the
“draping” model proposed by Dursi & Pfrommer (2008), for some angles to the line-of-sight. The
analysis of Sections 5.2, 5.4 and 5.5 suggests that the magnetic fields causing the RM bands are
well-ordered, consistent with a stretching of the initial field that has erased much of the small-scale
structure while amplifying the large-scale component. I next attempt to constrain the geometry of
the resulting “draped” field.
5.7.2 Axisymmetric draped magnetic fields
A proper calculation of the RM from a draped magnetic-field configuration (Dursi & Pfrommer
2008) is outside the scope of this work, but we can start to understand the field geometry using
some simple approximations in which the field lines are stretched along the source axis. I assumed
initially that the field around the lobe is axisymmetric, with components along and radially
outwards from the source axis, so that the RM pattern is independent of rotation about the axis. It
is important to stress that such an axisymmetric field is not physical, as it requires a monopole and
unnatural reversals, it is nevertheless a useful benchmark for features of the field geometry that are
needed to account for the observed RM structure. I first considered field lines which are parabolae
with a common vertex on the axis ahead of the lobe. For field strength and density both decreasing
away from the vertex, I found that RM structures, with iso-contours similar to arcs, rather than
bands, were generated only for the approaching lobe of an inclined source. Such anisotropic RM
structures were not produced in the receding lobes, nor for sources in the plane of the sky. Indeed,
in order to generate any narrow, transverse RM structures such as arcs or bands, the line-of-sight
must pass through a foreground region in which the field lines show significant curvature, which
occurs only for an approaching lobe in case of a parabolic field geometry. This suggested to
consider field lines which are families of ellipses centred on the lobe, again with field strength and
density decreasing away from the leading edge. This indeed produced RM structures in both lobes
for any inclination, but the iso-RM contours were arcs, not straight lines. Because of the nonphysical
nature of these axisymmetric field models, the resulting RM images are deliberately not
shown. In order to quantify the departures from straightness of the iso-RM contours, I measured
the ratio of the predicted RM values at the centre and edge of the lobe at constant Y, at different
96

5.7. Rotation-measure bands from a draped magnetic field 97
distances along the source axis, X, for both of the example axisymmetric field models. The ratio,
which is 1 for perfectly straight bands, varies from 2 to 3 in both cases, depending on distance
from the nucleus. This happens because the variations in line-of-sight field strength and density
do not compensate accurately for changes in path length. I believe that this problem is generic to
any axisymmetric field configuration.
The results of this section suggest that the field configuration required to generate straight RM
bands perpendicular to the projected lobe advance direction has systematic curvature in the field
lines (in order to produce a modulation in RM) without a significant dependence on azimuthal
angle around the source axis). I therefore investigated a structure in which elliptical field lines are
wrapped around the front of the lobe, but in a two-dimensional rather than a three-dimensional
configuration.
5.7.3 A two-dimensional draped magnetic field
I considered a field with a two-dimensional geometry, in which the field lines are families of
ellipses in planes of constant Y, as sketched in Fig. 5.12. The field structure is then independent
of Y. The limits of integration are given by the lobe surface and an ellipse whose major axis
is 4/3 of that of the lobe. Three example RM images are shown in Fig. 5.13. The assumed
density (2.4×10 −3 cm −3 ) and magnetic field (1µG) were the same as the pre-shock values for the
compression model of Section 5.6 and the lobe semi-major axis was again 21 kpc. Since the effect
of path length is very small (Section 5.6), the RM is also independent of Y to a good approximation.
The combination of elliptical field lines and invariance with Y allows us to produce straight RM
bands perpendicular to the projected lobe axis for any source inclination. Furthermore, this model
generates more significant reversals of the RM (e.g. Fig. 5.13c) than those obtainable with pure
compression (e.g. Fig. 5.11b).
I conclude that a field model of this generic type represents the simplest way to produce RM
bands with the observed characteristics in a way that does not require improbable initial conditions.
The invariance of the field with the Y coordinate is an essential point of this model, suggesting
that the physical process responsible for the draping and stretching of the field lines must act on
scales larger than the radio lobes in the Y direction.
5.7.4 RM reversals
The two-dimensional draped field illustrated in the previous section reproduces the geometry of
the observed RM bands very well, but can only generate a single reversal, which must be very
close to the front end of the approaching lobe, where the elliptical field lines bend most rapidly. A
97

98 5. Ordered magnetic fields around radio galaxies
Figure 5.12: Geometry of the two-dimensional draped model for the magnetic field, described
in Sec. 5.7.3. Panel (a): seen in the plane of the direction of the ambient magnetic field. Panel
(b): seen in the plane perpendicular to the ambient field; only the innermost family of ellipses is
plotted. For clarity, field lines behind the lobe are omitted. The colour of the curves indicates the
strength of the field, decreasing from red to yellow. The crosses represent the radio core position.
The coordinate system is the same as in Fig. 5.8 and the line of sight is in the X− Z plane.
IMAGE: BLANK.FITS
IMAGE: BLANK.FITS
IMAGE: BLANK.FITS
;100 ;50 0
RAD/M/M
rad m
0 10 20 30 40
RAD/M/M
rad m −2
;40 ;20 0 20
RAD/M/M
rad m
−2
x
x
x
(a) (b) (c)
Figure 5.13: Synthetic RM images produced for a two-dimensional draped model as described in
Section 5.7.3. The field lines are a family of ellipses draped around the lobe as in Fig. 5.12. Panel
(a): lobe inclined at 90 ◦ to the line-of-sight. (b) receding and (c) approaching lobes of a source
inclined by 40 ◦ to the line-of-sight. The crosses and the arrows represent the radio core position
and the lobe advance direction.
prominent reversal is apparent in the receding lobe of 0206+35 (Fig. 5.2a) and multiple reversals
across the eastern lobe of 3C 353 and in M 84 (Fig. 5.2b and d). The simplest way to reproduce
these is to assume that the draped field also has reversals, presumably originating from a more
complex initial field in the IGM. One realization of such a field configuration would be in the
form of multiple toroidal eddies with radii smaller than the lobe size, as sketched in Fig. 5.14.
Whatever the precise field geometry, I stress that the straightness of the observed multiple bands
again requires a two-dimensional structure, with little dependence on Y.
98

5.8. Discussion 99
5.8 Discussion
5.8.1 Where do bands occur?
The majority of published RM images of radio galaxies do not show bands or other kind of
anisotropic structure, but are characterized by isotropic and random RM distributions (e.g. Laing
et al. 2008, 3C 449, Chapter 4). On the other hand, I have presented observations of RM bands
in four radio galaxies embedded in different environments and with a range of jet inclinations
with respect to the line-of-sight. These sources are not drawn from a complete sample, so any
quantitative estimate of the incidence of bands is premature, but it is possible to draw some
preliminary conclusions.
The simple two-dimensional draped-field model developed in Section 5.7.3 only generates
RM bands when the line-of-sight intercepts the volume containing elliptical field lines, which
happens for a restricted range of rotation around the source axis. At other orientations, the RM
from this field configuration will be small and the observed RM may well be dominated by
material at larger distances which has not been affected by the radio source. I therefore expect
a minority of sources with this type of field structure to show RM bands and the remainder to have
weaker, and probably isotropic, RM fluctuations. In contrast, the three-dimensional draped field
model proposed by Dursi & Pfrommer (2008) predicts RM bands parallel to the source axis for a
significant range of viewing directions: these have not (yet) been observed.
The prominent RM bands here described occur only in lobed radio galaxies. In contrast, wellobserved
radio sources with tails and plumes seem to be free of bands or anisotropic RM structure
(e.g. 3C 31, 3C 449; Laing et al. 2008, Chapter 4). Furthermore, the lobes which show bands are all
quite round and show evidence for interaction with the surrounding IGM. It is particularly striking
that the bands in 3C 353 occur only in its eastern, rounded, lobe. The implication is that RM bands
occur when a lobe is being actively driven by a radio jet into a region of high IGM density. Plumes
and tails, on the other hand, are likely to be rising buoyantly in the group or cluster and I do not
expect significant compression, at least at large distances from the nucleus.
5.8.2 RM bands in other sources
The fact that RM bands have so far been observed only in a few radio sources may be a selection
effect: much RM analysis has been carried out for galaxy clusters, in which most of the sources
are tailed (e.g. Blanton et al. 2003). With a few exceptions like Cyg A (see below, Section 5.8.2),
lobed FR I and FR II sources have not been studied in detail.
99

100 5. Ordered magnetic fields around radio galaxies
Cygnus A
Cygnus A is a source in which we might expect to observed RM bands, by analogy with the
sources discussed in the present work: it has wide and round lobes and Chandra X-ray data have
shown the presence of shock-heated gas and cavities (Wilson, Smith & Young 2006). RM bands,
roughly perpendicular to the source axis, are indeed seen in both lobes (Dreher et al. 1987; Carilli
& Taylor 2002), but interpretation is complicated by the larger random RM fluctuations and the
strong depolarization in the eastern lobe. A semi-circular RM feature around one of the hot-spots
in the western lobe has been attributed to compression by the bow-shock (Carilli, Perley & Dreher
1988).
Hydra A
Carilli & Taylor (2002) have claimed evidence for RM bands in the northern lobe of Hydra A. The
Chandra image (McNamara et al. 2000) shows a clear cavity with sharp edges coincident with
the radio lobes and an absence of shock-heated gas, just as in the sources analysed in this work.
Despite the classification as a tailed source, it may well be that there is significant compression of
the IGM. Note, however, that the RM image is not well sampled close to the nucleus.
3C 465
The RM image of the tailed source 3C 465 published by Eilek & Owen (2002) shows some
evidence for bands, but the colour scale was deliberately chosen to highlight the difference between
positive and negative values, thus making it difficult to see the large gradients in RM expected at
band edges. The original RM image (Eilek, private communication) suggests that the band in
the western tail of 3C 465 is similar to those I have identified. It is plausible that magnetic-field
draping happens in wide-angle tail sources like 3C 465 as a result of bulk motion of the IGM
within the cluster potential well, as required to bend the tails. It will be interesting to search for
RM bands in other sources of this type and to find out whether there is any relation between the
iso-RM contours and the flow direction of the IGM.
5.8.3 Foreground isotropic field fluctuations
The coexistence in the RM images of the sources here analysed of anisotropic patterns with areas
of isotropic fluctuations suggests that the Faraday-rotating medium has at least two components:
one local to the source, where its motion significantly affects the surrounding medium, draping
the field, and the other from material on group or cluster scales which has not felt the effects
100

5.8. Discussion 101
line−of−sight
>
>
Figure 5.14: Geometry of a twodimensional
toroidal magnetic field as
described in Section 5.7.3, seen in the
plane normal to its axis. The cross
represents the radio core position.
of the source.
This raises the possibility that turbulence in the foreground Faraday rotating
medium might “wash out” RM bands, thereby making them impossible to detect. The isotropic
RM fluctuations observed across the sources are all described by quite flat power spectra with
low amplitude (Table 5.5). The random small-scale structure of the field along the line-of-sight
essentially averages out, and there is very little power on scales comparable with the bands. If, on
the other hand, the isotropic field had a steeper power spectrum with significant power on scales
similar to the bands, then its contribution might become dominant.
I first produced synthetic RM images for 0206+35 including a random component derived
from the best-fitting power spectrum (Table 5.5) in order to check that the bands remained visible.
I assumed a minimum scale of 2 kpc from the depolarization analysis for 0206+35 (Sec. 5.4), and a
maximum scale ofΛ max = 40 kpc, consistent with the continuing rise of the RM structure function
at the largest sampled separations, which requiresΛ max
> ∼ 30 arcsec (≃20 kpc). The final synthetic
RM is given by:
∫
RM syn = RM drap + RM icm =
n ′ e B′ z dz+ ∫
n e B z dz (5.6)
where RM drap and RM icm are the RM due to the draped and isotropic fields, respectively. The
terms n ′ e and B ′ z are respectively the density and field component along the line-of-sight in the
draped region. The integration limits of the term RM drap were defined by the surface of the lobe
and the draped region, while that of the term RM icm starts at the surface of the draped region and
extends to 3 times the core radius of the X-ray gas (Table 5.2). For the electron gas density n e
outside the draped region I assumed the beta-model profile of 0206+35 (Table 5.2) and for the
field strength a radial variation of the form (Laing et al. 2008, and references therein)
〈B 2 (r)〉 1/2 = B 0
[
ne (r)
n 0
] η
(5.7)
where B 0 is the rms magnetic field strength at the group centre. I took a draped magnetic field
strength of 1.8µG, in order to match the amplitudes for the RM bands in both lobes of 0206+35,
101

102 5. Ordered magnetic fields around radio galaxies
IMAGE: ELLISSOIDE_COUNTER_40_1.8_ISO.FITS
IMAGE: ELLISSOIDE_MAIN_40_1.8_ISO.FITS
;80 ;60 ;40 ;20 0 20 40 60
RAD/M/M
rad m −2
(a)
;80 ;60 ;40 ;20 0 20 40 60
RAD/M/M
rad m
−2
(b)
x
x
Receding
IMAGE: COUNTER_40_2.2_KOLMO.FITS
;50 0 50 100 150
RAD/M/M rad m −2
(c)
Approaching
IMAGE: FMATH.FITS
;100 ;50 0 50
RAD/M/M−2
rad m
(d)
x
x
Receding
Approaching
Figure 5.15: Synthetic RM images of the lobes of 0206+35, produced by the sum of a draped
field local to the source and an isotropic, random field spread through the group volume with the
best-fitting power spectrum found in Sec. 5.5 (panels (a) and (b)) and with a Kolmogorov power
spectrum (panels (c) and (d)). Both power spectra have the same high and low-frequency cut offs
and the central magnetic field strengths are also identical. The crosses and the arrows represent
the radio core position and the lobe advance direction.
and assumed the same value for B 0 .
Example RM images, shown in Figs. 5.15(a) and (b), should be compared with those for the
draped field alone (Figs. 5.13b and c, scaled up by a factor of 1.8 to account for the difference
in field strength) and with the observations (Fig.5.2a) after correction for Galactic foreground
(Table 5.4). The model is self-consistent: the flat power spectrum found for 0206+35 does not
give coherent RM structure which could interfere with the RM bands, which are still visible.
I then replaced the isotropic component with one having a Kolmogorov power spectrum
(q=11/3). I assumed identical minimum and maximum scales (2 and 40 kpc) as for the previous
power spectrum and took the same central field strength (B 0 =1.8µG) and radial variation (Eq. 5.7).
Example realizations are shown in Fig. 5.15(c) and (d). The bands are essentially invisible in the
presence of foreground RM fluctuations with a steep power spectrum out to scales larger than their
widths.
It may therefore be that the RM bands in the sources here presented are especially prominent
because the power spectra for the isotropic RM fluctuations have unusually low amplitudes and
102

5.8. Discussion 103
flat slopes. I have established these parameters directly for 0206+35, 3C 270 and 3C 353; M 84 is
only 14 kpc in size and is located far from the core of the Virgo cluster, so it is plausible that the
cluster contribution to its RM is small and constant.
I conclude that the detection of RM bands could be influenced by the relative amplitude and
scale of the fluctuations of the isotropic and random RM component compared with that from any
draped field, and that significant numbers of banded RM structures could be masked by isotropic
components with steep power spectra.
5.8.4 Asymmetries in RM bands
The well-established correlation between RM variance and/or depolarization and jet sidedness
observed in FR I and FR II radio sources is interpreted as an orientation effect: the lobe containing
the brighter jet is on the near side, and is seen through less magneto-ionic material (e.g. Laing
1988; Morganti et al. 1997). For sources showing RM bands, it is interesting to ask whether the
asymmetry is due to the bands or just to the isotropic component.
In 0206+35, whose jets are inclined by≈40 ◦ to the line-of-sight, the large negative band on
the receding side has the highest RM (Fig. 5.5). This might suggest that the RM asymmetry is due
to the bands, and therefore to the local draped field. Unfortunately, 0206+35 is the only source
displaying this kind of asymmetry. The other “inclined” source, M 84, (θ≈ 60 ◦ ) does not show
such asymmetry: on the contrary the RM amplitude is quite symmetrical. In this case, however,
the relation between the (well-constrained) inclination of the inner jets, and that of the lobes could
be complicated: both jets bend by≈90 ◦ at distances of about 50 arcsec from the nucleus (Laing et
al. in preparation), so that we cannot establish the real orientation of the lobes with respect to the
plane of the sky. The low values of the jet/counter-jet ratios in 3C 270 and 3C 353 suggest that their
axes are close to the plane of the sky, so that little orientation-dependent RM asymmetry would be
expected. Indeed, the lobes of 3C 270 show similar RM amplitudes, while the large asymmetry of
RM profile of 3C 353 is almost certainly due to a higher column density of thermal gas in front
of the eastern lobe. Within this small sample, there is therefore no convincing evidence for higher
RM amplitudes in the bands on the receding side, but neither can such an effect be ruled out.
In models in which an ordered field is draped around the radio lobes, the magnitude of any
RM asymmetry depends on the field geometry as well as the path length (cf. Laing et al. 2008
for the isotropic case). For instance, the case illustrated in Figs 5.13(b) and (c) shows very
little asymmetry even forθ= 40 ◦ . The presence of a systematic asymmetry in the banded RM
component could therefore be used to constrain the geometry.
103

104 5. Ordered magnetic fields around radio galaxies
5.8.5 Enhanced depolarization and mixing layers
A different mechanism for the generation of RM fluctuations was suggested by Bicknell, Cameron
& Gingold (1990). They argued that large-scale nonlinear surface waves could form on the surface
of a radio lobe through the merging of smaller waves generated by Kelvin-Helmholtz instabilities
and showed that RM’s of roughly the observed magnitude would be produced if a uniform field
inside the lobe was advected into the mixing layer. This mechanism is unlikely to be able to
generate large-scale bands, however: the predicted iso-RM contours are only straight over parts
of the lobe which are locally flat, even in the unlikely eventuality that a coherent surface wave
extends around the entire lobe.
The idea that a mixing layer generates high Faraday rotation may instead be relevant to the
anomalously high depolarizations associated with regions of compressed gas around the inner
radio lobes of M 84 and 3C 270 (Section 5.4 and Fig. 5.6). The fields responsible for the
depolarization must be tangled on small scales, since they produce depolarization without any
obvious effects on the large-scale Faraday rotation pattern. It is unclear whether the level of
turbulence within the shells of compressed gas is sufficient to amplify and tangle a pre-existing
field in the IGM to the level that it can produce the observed depolarization; a plausible alternative
is that the field originates within the radio lobe and mixes with the surrounding thermal gas.
5.9 Conclusions and outstanding questions
In this work I have analysed and interpreted the Faraday rotation across the lobed radio galaxies
0206+35, 3C 270, 3C 353 and M 84, located in environments ranging from a poor group to one
of the richest clusters of galaxies (the Virgo cluster). The RM images have been produced
at resolutions ranging from 1.2 to 5.5 arcsec FWHM using Very Large Array data at multiple
frequencies. All of the RM images show peculiar banded patterns across the radio lobes, implying
that the magnetic fields responsible for the Faraday rotation are anisotropic. The RM bands
coexist and contrast with areas of patchy and random fluctuations, whose power spectra have
been estimated using a structure-function technique. I have also analysed the variation of degree
of polarization with wavelength and compared this with the predictions for the best-fitting RM
power spectra in order to constrain the minimum scale of magnetic turbulence. I have investigated
the origin of the bands by making synthetic RM images using simple models of the interaction
between radio galaxies and the surrounding medium and have estimated the geometry and strength
of the magnetic field.
The results of this work can be summarized as follows.
104

5.9. Conclusions and outstanding questions 105
1. The lack of deviation fromλ 2 rotation over a wide range of polarization position angle and
the lack of associated depolarization together suggest that a foreground Faraday screen with
no mixing of radio-emitting and thermal electrons is responsible for the observed RM in the
bands and elsewhere (Section 5.3).
2. The dependence of the degree of polarization on wavelength is well fitted by a Burn law,
which is also consistent with (mostly resolved) pure foreground rotation (Section 5.4).
3. The RM bands are typically 3 – 10 kpc wide and have amplitudes of 10 – 50 rad m −2
(Table 5.4). The maximum deviations of RM from the Galactic values are observed at the
position of the bands. Iso-RM contours are orthogonal to the axes of the lobes. In several
cases, neighbouring bands have opposite signs compared with the Galactic value and the
line-of-sight field component must therefore reverse between them.
4. An analysis of the profiles of〈RM〉 and depolarization along the source axes suggests that
there is very little small-scale RM structure within the bands.
5. The lobes against which bands are seen have unusually small axial ratios (i.e. they appear
round in projection; Fig. 5.2). In one source (3C 353) the two lobes differ significantly in
axial ratio, and only the rounder one shows RM bands. This lobe is on the side of the source
for which the external gas density is higher.
6. Structure function and depolarization analyses show that flat power-law power spectra with
low amplitudes and high-frequency cut-offs are characteristic of the areas which show
isotropic and random RM fluctuations, but no bands (Section 5.5).
7. There is evidence for source-environment interactions, such as large-scale asymmetry
(3C 353) cavities and shells of swept-up and compressed material (M 84, 3C 270) in all
three sources for which high-resolution X-ray imaging is available.
8. Areas of strong depolarization are found around the edges of the radio lobes close to the
nuclei of 3C 270 and M 84. These are probably associated with shells of compressed hot
gas. The absence of large-scale changes in Faraday rotation in these features suggests that
the field must be tangled on small scales (Section 5.4).
9. The comparison of the amplitude of〈RM〉 with that of the structure functions at the largest
sampled separations is consistent with an amplification of the large scale magnetic field
component at the position of the bands.
10. I produced synthetic RM images from radio lobes expanding into an ambient medium
containing thermal material and magnetic field, first considering a pure compression of
105

106 5. Ordered magnetic fields around radio galaxies
both thermal density and field, and then including three- and two-dimensional stretching
(“draping”) of the field lines along the direction of the radio jets (Sects. 5.6 and 5.7). Both
of the mechanisms are able to generate anisotropic RM structure.
11. To reproduce the straightness of the iso-RM contours, a two-dimensional field structure
is needed. In particular, a two-dimensional draped field, whose lines are geometrically
described by a family of ellipses, and associated with compression, reproduces the RM
bands routinely for any inclination of the sources to the line-of-sight (Sec. 5.7.3). Moreover,
it might explain the high RM amplitude and low depolarization observed within the bands.
12. The invariance of the magnetic field along the axis perpendicular to the forward expansion
of the lobe suggests that the physical process responsible for the draping and stretching of
the magnetic field must act on scales larger than the lobe itself in this direction. It is not
possible to constrain yet the scale size along the line-of-sight.
13. In order to create RM bands with multiple reversals, more complex field geometries such as
two-dimensional eddies are needed (Section 5.7.4).
14. I have interpreted the observed RM’s as due to two magnetic field components: one draped
around the radio lobes to produce the RM bands, the other turbulent, spread throughout
the surrounding medium, unaffected by the radio source and responsible for the isotropic
and random RM fluctuations (Section 5.8.3). I tested this model for 0206+35, assuming
a typical variation of field strength with radius in the group atmosphere, and found that a
magnetic field with central strength of B 0 = 1.8µG reproduced the RM range quite well in
both lobes.
15. I have suggested two reasons for the low rate of detection of bands in published RM images:
our line of sight will only intercept a draped field structure in a minority of cases and rotation
by a foreground turbulent field with significant power on large scales may mask any banded
RM structure.
These results therefore suggest a more complex picture of the magneto-ionic environments
of radio galaxies than was apparent from earlier work. I find three distinct types of magneticfield
structure: an isotropic component with large-scale fluctuations, plausibly associated with the
undisturbed intergalactic medium; a well-ordered field draped around the leading edges of the
radio lobes and a field with small-scale fluctuations in the shells of compressed gas surrounding
the inner lobes, perhaps associated with a mixing layer. In addition, I have emphasised that simple
compression by the bow shock should lead to enhanced RM’s around the leading edges, but that
the observed patterns depend on the pre-shock field.
106

5.9. Conclusions and outstanding questions 107
MHD simulations should be able to address the formation of anisotropic magnetic-field
structures around radio lobes and to constrain the initial conditions.
107

108 5. Ordered magnetic fields around radio galaxies
108

Chapter 6
Faraday rotation in two extreme
environments ‡
In this Chapter I will present polarization analyses of two radio galaxies embedded in very
different environments. The sources are: B2 0755+37, located in a very sparse group, and
M 87, located at the centre of the cool core Virgo cluster. I will described the results of twodimensional
analyses of the RM and depolarization distributions. In both sources, the geometry
of the surrounding X-ray emitting gas appears to be complex than for the sources studied in
Chapters 4 and 5, so a three-dimensional analysis will be deferred to a later study.
6.1 The sources
The general parameters of the sources are listed in Table 6.1, while Table 6.2 shows the X-ray
parameters taken from the literature and the equipartition parameters derived from radio data in
this work. I refer to parts of the sources by the abbreviations N, S, E, W (for North, South, East,
West).
6.1.1 B2 0755+37
B2 0755+37 (hereafter 0755+37) is an extended FR I radio source whose optical counterpart,
NGC 2484, is a D-galaxy, classified as central member of a nearby and very poor group of galaxies
(Mulchaey et al. 2003). The source is one of the most isolated radio galaxies in the B2 catalogue.
Here I briefly summarize the radio properties of 0755+37, which are described in detail in the
Appendix of this thesis along with a description of the VLA observations. At a resolution of
‡ Guidetti et al. in prep
109

110 6. Faraday rotation in two extreme environments
1.3 arcsec the source shows a very bright core and two-sided jets embedded in an diffuse halo of
low surface brightness. The jet E of the nucleus is the brighter one and the jets inclination to the
line-of-sight is estimated to be about 30 ◦ (Bondi et al. 2000; Laing & Bridle in preparation) The
radio source is highly polarized.
0755+35 has been observed with both the PSPC and HRI instruments on ROSAT (Worrall &
Birkinshaw 1994, 2000) and with Chandra (Worrall, Birkinshaw & Hardcastle 2001).
The X-ray emission has extended thermal and partially resolved non-thermal components,
respectively associated with the hot medium on group and galactic scales, and with the core
and jets (Canosa et al. 1999; Worrall & Birkinshaw 2000; Mulchaey et al. 2003; Wu, Feng &
Xinwu 2007). However, the X-ray surface brightness profile (the sum of aβmodel and a point
component) is very poorly constrained. The extent of the thermal X-ray emission is relatively
small with respect to other poor groups, particularly along the EW axis of the large-scale source
structure, and its luminosity is low (Worrall & Birkinshaw 2000). These arguments together with
the “isolation” of 0755+37 suggest that the diffuse emission, at least on the scale of the radio
emission, is mainly due to the hot atmosphere surrounding the galaxy.
6.1.2 M 87
M 87 is one of the most famous radio galaxies and resides in the gas-rich Virgo cluster, which is the
nearest, bright X-ray emitting cluster. This source provides the opportunity to study the magnetic
field fluctuations with excellent spatial resolution. It is a popular target of multiwavelength studies,
because of its one-sided jet observed at radio (e.g. Owen 1989 (VLA); Kovalev 2007 (VLBA),
optical Biretta, Sparks & Macchetto 1999) and X-ray frequencies (e.g. Forman et al. 2005, 2007;
Harris et al. 2006; Churazov 2008). The morphology of the jet is similar at all these wavelengths,
suggesting a common synchrotron radiation mechanism.
M 87 is classified as an FR I source. On scales larger than the radio jet (∼2 kpc), it has a
complex radio structure mapped out to linear scales of about 40 kpc, showing inner and outer
lobes, curved “ears”, and filaments (Fig. 6.2, Owen, Eilek & Kassim 2000). The jet inclination is
estimated to be about 22 ◦ (Biretta, Zhou & Owen 1995).
In this work, I used VLA datasets at 5 and 8 GHz (6 cm and 3 cm respectively), which were
generously provided by F. N.Owen. The VLA observations in the 6 cm band and their reduction
were presented by Owen, Eilek & Keel (1990) while the 3 cm data have not been published. At
these frequencies and at 0.4 arcsec resolution the source is highly polarized. The images show the
main jet, pointing to NW, and two “inner lobes” extending about 4 kpc from the core (Owen, Eilek
& Keel 1990).
110

6.2. Two-dimensional analysis: rotation measure and depolarization 111
Table 6.1: General optical and radio properties: Col. 1: source name; Cols. 2, 3: position; Col.
4: redshift; Col. 5: conversion from angular to spatial scale with the adopted cosmology; Col. 6:
Fanaroff-Riley class; Col. 7: the largest angular size of the radio source; Col. 8: radio power at
1.4 GHz; Col. 9: angle to the line of sight of the jet axis.
source RA DEC z kpc/arcsec FR class LAS θ
[J2000] [J2000] [arcsec] [degree]
0755+37 07 58 28.1 +37 47 12 0.0428 0.833 I 139 30
M 87 12 30 49.4 +12 23 28 0.00436 0.089 I 489 22
Table 6.2: X-ray and radio equipartition parameters for all the sources. Col. 1: source name; Col.
2: X-ray energy band; Col. 3: average thermal temperature; Cols. 4,5,6 and 7,8,9 best-fitting core
radii, central densities andβparameters for the outer and innerβmodels, respectively; Col. 10:
average thermal pressure at the midpoint of the radio lobes; Cols. 11&12: minimum synchrotron
pressure and corresponding magnetic field; Col.13: references for the X-ray models.
source band kT r cxout n 0out β out r cxin n 0in β in P 0 P syn B Psyn ref.
[keV] [keV] [kpc] [cm −3 ] [kpc] [cm −3 ] [dyne cm −2 ] [dyne cm −2 ] [µG]
0755+37 0.2-2.5 0.73 +0.32
−0.45
159 6.4×10 −4 0.9 1.7×10 −12 3.14×10 −13 4.4 1
M 87 0.7-2.7 1.68 0.04
−0.05 23.4 0.011 0.47 1.86 0.13 0.42 2.7×10−10 2.11×10 −10 55 2
References: (1) Worrall & Birkinshaw (2000); (2) Matsushita et al. (2002).
Fields in the Virgo cluster centred on M 87 have been observed by several X-ray satellites.
The extent of the X-ray emission is larger than 2 ◦ , which is the field of view of the ROSAT PSPC,
and is centrally peaked, implying that M 87 harbours a cool core.
At the ROSAT and XMM-Newton resolutions, the morphology of the X-ray emission appears
regular and smooth, but deep Chandra observations (e.g. Forman et al. 2007 and references therein)
have revealed a very complex structure suggesting interaction between the radio source and the
hot gas on several spatial scales (Fig. 6.3). In particular, Forman et al. (2007) detected multiple
cavities, some of which coincide with the inner radio lobes, rims of enhanced X-ray emission,
loops, rings, and filaments. The cavities resemble the outer large-scale radio structure and have
been interpreted as a system of buoyant bubbles produced in AGN outbursts. At distances of about
10-20 kpc from the core, temperature and density jumps are apparent. These correspond to weak
shocks with Mach numbersM≈1.4.
6.2 Two-dimensional analysis: rotation measure and depolarization
I produced RM images and related rms errors by weighted least-squares fitting to the polarization
angle mapsΨ(λ) as a function ofλ 2 (Eq. 3.1) at respectively three and eight frequencies for
0755+37 and M 87 (Table 6.3) by using a version of the RM task in the AIPS package modified
111

112 6. Faraday rotation in two extreme environments
IMAGE: 07_PSPC_CONVL_AS_1.3_OGEOM.FITS
CNTRS: 07_1.3_I_4.FITS
0 0.0002 0.0004 0.0006 0.0008
37 49 00
50 kpc
DECLINATION (J2000)
37 48 00
37 47 00
37 46 00
37 45 00
07 58 40
07 58 30
RIGHT ASCENSION (J2000)
07 58 20
Figure 6.1: Overlay of the X-ray emission (ROSAT PSPC) around 0755+37 on radio contours at
1.385 GHz at 4.0 arcsec resolution.
Table 6.3: Frequencies, bandwidths and angular resolutions used in the RM and Burn law k
images. Details of the 0755+37 dataset are given in the Appendix.
source ν ∆ν beam
[MHz] [MHz] [arcsec]
0755+37 1385.1 12.5 1.3-4.0
1464.9
4860.1 100
M 87 4635.1 50 0.4
4735.0
4835.1
4935.0
8085.1
8335.1
8535.1
8785.1
by G.B. Taylor. The RM maps were calculated only at pixels with polarization angle uncertainties
≤10 ◦ at all frequencies. The RM maps have not been corrected for the Galactic contribution. For
0755+37, this is estimated to be negligible (-1±2 rad m −2 ; Simard-Normandin et al. 1981). For
M 87, the value is less well-determined, but the source is at very high Galactic latitude, and the
associated RM must be very small compared with the measured values (see below). Values of
σ RM integrated over the maps are not corrected for the fitting errorσ RMfit ,
I produced Burn law k images at the same resolutions and using the same frequencies as the
RM maps. Weighted least-squares fits were made to the fractional polarization maps ln p(λ) as a
112

6.2. Two-dimensional analysis: rotation measure and depolarization 113
Figure 6.2: Composite radio image of M87 showing the complex structure on multiple spatial
scales and at several frequencies. In the present work, the RM from the central cone-like structure,
shown in more detail in the top-left panel, has been analysed. Image courtesy of NRAO/AUI.
ROSAT XMM−Newton Chandra
Figure 6.3: X-ray images of the centre of the Virgo cluster showing different scales: ROSAT
(left) XMM-Newton (middle) Chandra (right). The increasing resolution has shown a complex
structure in the hot gas. Courtesy of W. Forman.
function ofλ 4 (Eq. 3.5), using a modified version of AIPS RM task. Only data with p>4σ p at
each frequency were included in the fits.
6.2.1 0755+37: images
Two RM images have been produced, at resolutions of 1.3 and 4.0 arcsec. Both of the fits are very
good, showing no evidence for deviations fromλ 2 rotation, although the total range of rotation is
quite small. The RM maps are both shown in Fig. 6.4.
113

114 6. Faraday rotation in two extreme environments
At 4.0 arcsec the RM distribution has a mean of−5.4 rad m −2 withσ RM =5.1 rad m −2 and
an average fitting errorσ RMfit
σ RM =6.0 rad m −2 andσ RMfit = 2.0 rad m −2 .
= 0.8 rad m −2 . At 1.3 arcsec the mean RM is−0.1 rad m −2 with
The small amplitude of the RM fluctuations is consistent with an origin in the IGM of the
isolated host galaxy, rather than the atmosphere of a more extended group.
There is a clear asymmetry in RM structure between the E and W lobes at both resolutions.
Firstly, most of the E (approaching) lobe is characterized by a fairly uniform, positive RM, and
the largest RM fluctuations, of both signs, are seen in the W (receding) lobe. Secondly, both lobes
show anisotropic features, but with different characteristics. In the E lobe there is a thin, straight
structure elongated parallel to the source axis with almost zero〈RM〉 . This “stripe” is very striking
on the 1.3 arcsec map and appears to be almost coincident with the main jet axis, extending beyond
the apparent termination of the jet as far as the boundary of the lobe. It is probably the clearest
known example of kpc-scale, jet-related RM structure. It can be interpreted in at least two possible
ways:
1. the jet is mostly in front of the Faraday screen. This might happen if the jet has displaced a
relatively large amount of hot gas or if it is bending along the line-of-sight.
2. The jet lies behind a Faraday screen with an intrinsically low RM along the stripe, i.e., the
magnetic field is feeble and/or preferentially aligned in the plane of the sky on the projected
jet axis.
The latter point requires that the Faraday screen somehow knows about the jet.
The 4.0-arcsec RM map reveals arc-like features at the leading edge of the W lobe. These have
alternating signs and must therefore be associated with field reversals. They are confirmed by the
higher resolution map, where the iso-RM contours appear to be fairly straight and perpendicular
to the jet axis. The resemblance to the RM bands discussed in Chapter 5 is obvious. It is possible
that these features are also caused by draping of the magnetic field around the leading edge of the
lobe.
The Burn law k maps are shown in Fig. 6.5. At 4.0 arcsec resolution, the mean value of k is
106 rad 2 m −4 , corresponding to a depolarization DP 21cm
6cm
= 0.79. The mean observed k values for
sub-regions of 0755+37 at both the angular resolutions are listed in Table 6.4. From this table, but
also by visual inspection of Fig. 6.5 (left), it is evident that at 4.0 arcsec the highest depolarization
is observed in the W (receding) lobe. This is consistent with a higher path length through the X-ray
emitting gas (Laing-Garrington effect; Laing 1988; Garrington et al. 1988). In contrast, the Burn
law k distribution at 1.3 arcsec is very symmetrical and has a much lower overall mean, implying
that the RM fluctuations are mostly resolved, consistent with a foreground screen.
114

6.2. Two-dimensional analysis: rotation measure and depolarization 115
IMAGE: 0755.RM.4.0_BLANK.FITS
DECLINATION (J2000)
37 48 00
37 47 30
37 47 00
37 46 30
;20 ;10 0 10 20
RAD/M/M
07 58 36
07 58 32 07 58 28 07 58 24
RIGHT ASCENSION (J2000)
10 kpc
DECLINATION (J2000)
37 48 00
37 47 30
37 47 00
37 46 30
IMAGE: 0755.RM.1.3_BLANK.FITS
;20 ;10 0 10 20
RAD/M/M
07 58 36
07 58 32 07 58 28 07 58 24
RIGHT ASCENSION (J2000)
10 kpc
07 58 20
Figure 6.4: Images of the rotation measure for 0755+37, computed from weighted least-squares
fits to the position angle –λ 2 relation at frequencies of 1385.1, 1469.1 and 4860.1 MHz. The
angular resolutions are 4.0 arcsec FWHM (left) and 1.3 arcsec FWHM (right).
IMAGE: 0755_K;BURN_4ARCSEC_4SIGMAP.FITS
0 100 200 300 400
rad^2/m^4
IMAGE: 0755_K;BURN_1.3ARCSEC.FITS
0 100 200 300 400
rad^2/m^4
37 48 00
37 48 00
DECLINATION (J2000)
37 47 30
37 47 00
37 46 30
DECLINATION (J2000)
37 47 30
37 47 00
37 46 30
07 58 36
07 58 32 07 58 28 07 58 24
RIGHT ASCENSION (J2000)
07 58 36
07 58 32 07 58 28 07 58 24
RIGHT ASCENSION (J2000)
07 58 20
Figure 6.5: Images of Burn law k for 0755+37 computed from weighted least-squares fits to the
relation p(λ)= p(0)exp(−kλ 4 ) at frequencies of 1385.1, 1469.1 and 4860.1 MHz. The angular
resolutions are 4.0 arcsec FWHM (left) and 1.3 arcsec FWHM (right).
This scenario is confirmed by the profiles of〈k〉 across the source (Fig. 6.6): at lower resolution
the profile is asymmetrical, with less depolarization at a given distance from the core in the E lobe
and a peak which is displaced slightly W of the nucleus. In contrast, the〈k〉 profile at 1.3 arcsec
resolution is symmetrical and lies significantly below that at 4.0 arcsec.
The high k values in the W lobe at the lower resolution are mostly seen in a cone-like structure
(Fig. 6.5 left), as was already pointed out by Bondi et al. (2000). It is possible that the origin of
such structure may be gas and field compression or the development of a mixing layer as discussed
for M 84 and 3C 270 (Sec. 5.8.5). To confirm this picture, higher resolution X-ray observations are
needed. In contrast, a region with very low and almost constant depolarization is observed at the
leading edge of the W lobe, beyond the end of the approaching jet.
115

116 6. Faraday rotation in two extreme environments
0755+37
Figure 6.6: Profiles of Burn law k for 0755+37 parallel to the jet direction, plotted against distance
from the nucleus. The angular resolutions are 4.0 (left) and 1.3 arcsec FWHM (right).
Apart from the cone-like structure in the E lobe no correlation of depolarization with source
morphology is apparent. Together with the absence of significant deviations fromλ 2 rotation, this
is again consistent with a Faraday rotating medium mostly external to the source.
6.2.2 M 87: images
I made the RM image of M 87 at a resolution of 0.4 arcsec. TheRM task did not solve correctly for
the nπ ambiguities in position angle at some locations. In order to remove these ambiguities, the
RM map was produced using the following steps:
1. Fit to the four frequencies in the 3 cm band using the Greg Taylor version of theRM program
inAIPS. This produced a noisy, but smooth RM image.
2. RunMWFLT inAIPS on this RM image, to apply a median-weight filter with a 15-pixel
(1.5 arcsec) smoothing kernel.
3. Correct all of the polarization angle maps (in the 6 and 3 cm bands) using the filtered RM
image.
4. Run a modified version ofRM task which minimises the absolute differences between
the polarization angle maps at adjacent frequencies (i.e. assumes that there are no nπ
ambiguities).
5. Add the result back to the filtered RM image.
The resulting RM map essentially confirms the results of the 4-frequency, C-band fit made by
Owen, Eilek & Keel (1990), but is significant improved both in noise level and overall reliability.
116

6.2. Two-dimensional analysis: rotation measure and depolarization 117
IMAGE: M87.RMRAL_BLANK.FITS
DECLINATION (J2000)
12 23 45
12 23 30
12 23 15
12 23 00
;2000 0 2000 4000 6000
RAD/M/M
1 kpc
12 22 45
12 30 52
12 30 51
12 30 50 12 30 49 12 30 48
RIGHT ASCENSION (J2000)
12 30 47
Figure 6.7: Rotation measure of M 87 computed from fits to the position angle –λ 2 rotation at
eight frequencies as described in the text. The angular resolution is 0.4 arcsec FWHM.
The new map, shown in Fig. 6.7, has an average fitting errorσ RMfit = 15 rad m −2 . The RM
distribution has a mean of roughly 650 rad m −2 withσ RM ≈900 rad m −2 .
The main features of the RM image are as follows.
• The RM values are large and almost always positive, requiring a magnetic field pointing
towards us across most of the source. They range from≈−2000 up to≈6000 rad m −2 . This
compares with the much larger range of−1000-8000 rad m −2 found by Owen, Eilek & Keel
(1990). The most extreme positive values found by are Owen, Eilek & Keel (1990) not
consistent with the X-band position angles.
• The highest RM values (≈3500-6000 rad m −2 ) are seen only in a restricted patch in the E
lobe.
• The jet appears as a uniform RM structure characterized by low values (≈100 rad m −2 ), as
already pointed out by Owen, Eilek & Keel (1990).
• The RM distribution is asymmetrical: positive and smoothly varying in the W lobe, patchier
and associate with field reversals in the E lobe.
As in 0755+37, the low RM seen across the jet is surprising. Similar explanations may hold
for the two sources.
117

118 6. Faraday rotation in two extreme environments
0 20000 40000 60000
12 24 00
12 23 40
DECLINATION (J2000)
12 23 20
12 23 00
12 22 40
12 30 51
12 30 50
12 30 49
12 30 48
12 30 47
RIGHT ASCENSION (J2000)
Figure 6.8: Left: Image of Burn law k for M 87 computed from weighted least-squares fits to the
relation p(λ)= p(0)exp(−kλ 4 ) at the eight frequencies listed in Table 6.3. Right: Profiles of Burn
law k along the jet direction, plotted against distance from the nucleus. The angular resolution is
0.4 arcsec FWHM.
The Burn law k map for M 87 is shown in Fig. 6.8. At 0.4 arcsec resolution the overall k
distribution has a mean of 9820 rad 2 m −4 , corresponding to a depolarization DP 6cm
3cm
= 0.85. The
mean observed k values for the two lobes are listed in Table 6.4. The highest depolarization occurs
in patchy structures in the E (receding) lobe, consistent with the Laing-Garrington effect (Laing
1988; Garrington et al. 1988). These structures appear filamentary and are likely to be caused by
steep (and therefore unresolved) RM gradients. Indeed, they are in the same areas as the highest
absolute values of RM and∇RM (Fig. 6.7).
In addition to low and constant RM, the jet also shows uniformly low k.
The profile of
〈k〉 across the source (Fig. 6.8, right) is qualitatively similar to that observed in 0755+37, but
more extreme: it increases monotonically from W to E across the core, peaks in the E lobe
and then decreases at larger distances. Except in the region of the jet, no detailed correlation
of depolarization with source morphology is apparent. Together with the absence of significant
deviations fromλ 2 rotation, this again requires a Faraday rotating medium mostly external to the
source.
6.3 Two-dimensional analysis: structure functions
I used the RM structure function (Eq. 3.13) to quantify the two-dimensional fluctuations of Faraday
rotation measure on scales larger than the observing beamwidth for 0755+37 and M 87.
118

6.3. Two-dimensional analysis: structure functions 119
0755+37
Figure 6.9: RM structure functions for the sub-regions of 0755+37 indicated in the panels at 4.0
and 1.3 arcsec FWHM. The horizontal bars represent the bin widths and the crosses the mean
separation for data included in the bins. The red curves are the best fits, including the effects of
the convolving beam. The error bars represent the rms variations for structure functions derived
from multiple realizations of the indicated power spectrum on the observed grid.
119

120 6. Faraday rotation in two extreme environments
Figure 6.10: RM structure functions for the sub-regions of M 87 indicated in the panels at
0.4 arcsec FWHM. The horizontal bars represent the bin widths and the crosses the mean
separation for data included in the bins. The red curves are the best fits including the effects
of the convolving beam. The error bars represent the rms variations for structure functions derived
for multiple realizations of the indicated cut-off power-law power spectra on the observed grid for
each region.
Structure functions have been computed at the resolutions of the RM images and binned by
separation over four areas of 0755+37 (the two jets and the two lobes), and in three areas of M 87
(the inner and outer E lobe and the W lobe). The reason for splitting the E lobe of M 87 into two
sub-regions was that the structure function derived for the whole lobe showed a decrease at large
separation which probably indicates the presence of significant spatial variations in the field and/or
density (cf. 3C 449, Chapter 4). All of these sub-regions have enough data points to ensure that
the structure functions are robustly determined. A first-order correction for uncorrelated random
noise,σ fit , evaluated in each sub-region, has been applied to all of the structure functions. A
cut-off power law (CPL) power spectrum (see also Eq. 4.3)
Ĉ( f ⊥ ) = 0 f ⊥ < f min
= C 0 f −q
⊥
f ⊥ ≤ f max
= 0 f ⊥ > f max . (6.1)
gives model structure functions in very good agreement with the data for both sources (Figs.6.9
and 6.10). The values of f max have also been estimated by fitting the mean values of k for
the sub-regions. I produced multiple RM realisations corresponding to the CPL model power
spectrum on the observed grid, including the effects of the convolving beam, to evaluate the rms
deviations of their structure functions. These take into account the effects of undersampling and
are plotted as error bars attached to the observed points. The best-fitting parameters are quoted
for each sub-region separately in Table 6.4, together with the observed and predicted values of
120

6.4. Preliminary conclusions 121
Table 6.4: Best-fitting power spectrum parameters for the individual sub-regions of 0755+37 and
M 87.
Source Region FWHM q f max f min k obs k syn
[arcsec] [arcsec −1 ] [arcsec −1 ] [rad 2 m −4 ] [rad 2 m −4 ]
0755+37 E LOBE 4.0 2.48 8.33 ∞ 64±3 69±3
E JET 2.45 ” 0.017 112±5 119±4
W JET 2.20 ” 0.052 156±7 164±5
W LOBE 2.19 0.69 0.028 109±5 104±5
E LOBE 1.3 2.52 8.33 ∞ 41±3 37±4
E JET 2.51 4.16 0.017 63±4 57±5
W JET 2.21 8.33 0.052 58±5 55±5
W LOBE 2.19 0.69 0.028 38±6 40±4
M 87 E out LOBE 0.4 3.05 2.6 ∞ 23470±600 34350±150
E in LOBE ” 2.87 2.6 ” 35310±800 59540±200
W LOBE ” 2.85 2.6 ” 15178±500 14491±100
Burn law k. In M 87, the slopes and cut-off frequencies derived for the individual sub-regions are
mutually consistent. A combined fit, weighting all of the sub-regions equally and allowing their
normalizations to vary, gives q=2.95 and f max = 2.75 arcsec −1 . The slopes for the different parts
of 0755+37 differ significantly from each other, and no combined fit has been attempted.
The structure functions for the lobes of M 87 show little firm evidence for any levelling off at
large separations (the points at> ∼ 15 arcsec in the East lobe have very large errors). In the W lobe,
the field outer scale must be> ∼ 30 arcsec (≃2.7 kpc). The fits therefore assume f min =∞ for this
source. In contrast, the structure functions for both of the jets and the W lobe of 0755+37 level
off and clearly show the suppression of the RM fluctuations above some outer scale. Better fits
to these structure functions are provided by setting a finite value for f min of 0.017-0.052 arcsec −1 ,
corresponding to a maximum field fluctuations on scalesΛ max ≈ 15−50 kpc. The situation is
more ambiguous in the E lobe of 0755+37, because of the large errors, and in this case the fit
with f min =∞ is shown. Further work on the outer scale will require a three-dimensional analysis
which takes into account the spatial variations of density and path length across the sources.
6.4 Preliminary conclusions
I have studied in detail the images of linearly polarized emission from the FR I radio galaxies
0755+37, in a sparse group, and M 87 at the centre of the cool-core Virgo cluster. I have produced
images of RM and Burn law k with resolutions of 4.0 and 1.3 arcsec FWHM for 0755+37, and
0.4 arcsec FWHM for M 87. The spatial variations of the RM distributions in both sources have
been investigated using a structure-function technique.
121

122 6. Faraday rotation in two extreme environments
The results of this analysis can be summarized as follows:
• The polarization position angles accurately follow theλ 2 relation. Together with the
observed low depolarization, this confirms yet again that the Faraday rotation is primarily
in front of the radio emission. The residual depolarization is consistent with unresolved RM
fluctuations.
• The amplitude of the RM fluctuations in 0755+37 is smaller than for FR I sources embedded
in groups of galaxies (Chapter 5) and comparable to that in NGC 315, which is also in a
very sparse environment (Laing et al. 2006a). This reinforces the hypothesis the most of the
Faraday rotating material is due to the hot atmosphere of the associated galaxy.
• In M 87 the amplitude of the RM fluctuations is typical of that found in other cool-core
clusters such as Cygnus A and Hydra A (e.g. Carilli & Taylor 2002; Laing et al. 2008).
• The profiles ofσ RM (which are not shown) and〈k〉 for both sources have higher average
values for the receding lobes. Thus both resolved and sub-beam fluctuations of RM are
higher on the receding side (the Laing-Garrington effect; Laing 1988; Garrington et al.
1988).
• Both approaching jets show low and uniform rotation measures. In M 87, the associated
depolarization is also anomalously low.
• There is other evidence for anisotropic RM fluctuations in 0755+37: the leading edge of
the W lobe shows arc-like RM structures with sign reversals. These are reminiscent of the
RM bands discussed in Chapter 5. The RM structures of both 0755+37 and M 87 appear
reasonably isotropic elsewhere.
• Except in the restricted areas showing obvious anisotropic structure, it is reasonable to
treat the RM as a Gaussian, random, isotropic variable and to estimate its power spectrum.
The power spectra are well described by power laws with slopes ranging from 2.2 to 2.5
in 0755+37 and slightly steeper (≈3.0) in M 87. These are comparable with the slopes
deduced for 3C 449, 3C 31 (Laing et al. 2008) and the regions not dominated by RM bands
in 0206+35, 3C 270 and 3C 353 (Chapter 5). They are all significantly flatter than the value
of 11/3 expected for Kolmogorov turbulence.
• These power spectra are consistent with the observed depolarization for minimum field
scales of 0.1 kpc in 0755+37 and 0.06-0.25 kpc in M 87. The nearby location and high
surface brightness of M 87 have allowed an accurate determination of its RM power
spectrum on scales smaller than those accessible for other objects.
122

6.4. Preliminary conclusions 123
• The derived magnetic-field auto-correlation lengths λ B are 0.1-1.4 kpc in 0755+37
(depending on the sub-region), and 0.2 kpc in M 87.
• The central magnetic field strength required to match the RM fluctuations, deduced from
the single-scale approximation including cavities and assuming these values ofλ B as scalelengths,
is≈1.0µG in 0755+37 and≈35µG in M 87. This confirms the trend for stronger
fields to be seen in richer environments (Carilli & Taylor 2002), although subject to
confirmation by three-dimensional modelling.
Significant imprecision is certainly introduced by the assumption of constant integration path
required to interpret the RM structure functions over limited sub-regions. This can be corrected
in future three-dimensional simulations, but better constraints on the source geometries and the
distributions of hot gas will be required. This will be particularly challenging for M 87, where the
hot-gas structure is extremely complicated (Forman et al. 2007).
123

124 6. Faraday rotation in two extreme environments
124

Summary and conclusions
The goal of this thesis was to constrain the strength and structure of the magnetic field associated
with the extragalactic medium surrounding radio galaxies located in a variety of environments by
using observations of Faraday rotation and depolarization. To interpret the RM structure, I have
derived analytical models and performed two- and three-dimensional Monte Carlo simulations.
The picture that emerges is that the magneto-ionic environments of radio galaxies are
significantly more complicated than was apparent from earlier work. The unique feature of this
thesis is that the target radio galaxies are all large and highly polarized, enabling the determination
of Faraday rotation and depolarization at a large number of independent points with high signalto-noise
ratio. This in turn allows global variations across the sources to be determined, as well as
accurate estimates of spatial statistics.
In Section 6.5, I summarize the main results obtained in Chapters 4, 5 and 6. The general
conclusions are given in Sec. 6.6 as a set of answers to the questions posed in the Abstract and
finally Sec. 6.7 briefly lists some topics for further work.
6.5 Summary
6.5.1 The tailed source 3C 449
The RM and depolarization across the source both appear to be consistent with a pure, mostly
resolved foreground Faraday screen. The RM structure shows no preferred direction anywhere
in the radio source. This is consistent with an isotropic, turbulent magnetic field fluctuating over
a wide range of spatial scales. I quantified the statistics of the magnetic-field fluctuations by
deriving RM structure functions, which have been fitted using models derived from theoretical
power spectra. The errors due to undersampling have been estimated by making multiple twodimensional
realizations of the best-fitting power spectrum. Depolarization measurements have
also been used to estimate the minimum scale of the magnetic field variations. I then developed
three-dimensional models with a gas density distribution derived from X-ray observations and
125

126 6. Summary and conclusions
a random magnetic field with the previously-determined power spectrum. By comparing the
simulations with the observed Faraday rotation images, the strength of the magnetic field and
its dependence on density could be estimated. The RM and depolarization data are consistent with
a broken power-law magnetic-field power spectrum, with a break at about 11 kpc and slopes of
2.98 and 2.07 at smaller and larger scales, respectively. The minimum scale of the fluctuations is
≈0.2 kpc. A particularly interesting result is the determination of the outer scale,Λ max ≈65 kpc,
which gives the driving scale of turbulence. This is the first time that this quantity has been
estimated unambiguosly.
The average magnetic field strength at the group centre is 3.5±1.2µG, decreasing linearly
with the gas density within≈16 kpc of the nucleus. At larger distances, the dependence of field
on density appears to flatten, but this may be an effect of errors in the density model. The ratio of
the thermal and magnetic-field pressures is≈30 at the nucleus and≈400 at the core radius of the
group gas. Therefore, the intra-group magnetic field is not energetically important.
These results indicate a magnetized foreground medium very similar to that in 3C 31 (Laing et
al. 2008), as was expected from the close similarity between the environments, radio morphologies
and redshifts of the two sources. The RM fluctuation amplitudes are comparable, as is the derived
central magnetic field (B 0 ≈ 2.8µG for 3C 31) and the magnetic field power spectrum which
in both sources can be fit by a broken power law steepening at higher spatial frequencies. The
low-frequency slopes are similar (q=2.3 for 3C 31), but the high-frequency slope for 3C 31 is
consistent with the value for Kolmogorov turbulence (q=11/3). Because of uncertainties in the
density distribution around 3C 31, Laing et al. (2008) could only estimate a lower limit to the outer
scale of magnetic-field fluctuations,Λ max
> ∼ 70 kpc, likely set as in 3C 449 by interactions with
companion galaxies in the group.
6.5.2 The lobed radio galaxies 0206+35, 3C 270, M 84 and 3C 353
In contrast to 3C 449, all of the sources show highly anisotropic banded RM structures with
contours of constant RM perpendicular to the major axes of their radio lobes. The bands have
widths of 3-10 kpc and amplitudes of 10-50 rad m −2 . In several cases, they are associated with
field reversals. All of the sources except M 84 also have regions in which the RM fluctuations
have lower amplitude and appear isotropic. Structure function and depolarization analyses for
these areas have shown that flat power-law power spectra with low amplitudes and high-frequency
cut-offs give good descriptions of the spatial statistics. The strength of the field which gives
rise to the bands is significantly higher than that of the largest-scale component in the band-free
regions. The wavelength behavior of the polarization position angles and the almost complete
absence of depolarization imply, as usual, that a mostly resolved foreground Faraday screen is
126

6.5. Summary 127
responsible for the observed RM in the bands and elsewhere. I have presented an initial attempt to
interpret the banded RM phenomenon as a consequence of interactions between the sources and
their surroundings. Synthetic RM images have been produced for the case of compression of a
uniformly-magnetized external medium. This simple model of the source-environment interaction
predicts RM bands of approximately the right amplitude, but only under special initial conditions.
A much better description of the data is provided by a two-dimensional magnetic structure in
which the field lines are a family of ellipses draped around the leading edge of the lobe. This
draped-field model can produce RM bands in the correct orientation for any source inclination.
Moreover, it might explain the high RM amplitude and low depolarization observed within the
bands. The two-dimensional draped-field model may also account for the low rate of detection of
RM bands in published observations, since anisotropic RM structures will not be observed unless
the line-of-sight intercepts the wrapped field lines. Furthermore, superposed RM fluctuations with
significant power on scales comparable with the bands will make them difficult to recognise.
This work has demonstrated unequivocally for the first time that the interaction between radio
galaxies and their immediate environments affects the magnetisation of the surrounding plasma.
Finally, I have reported rims of high depolarization at the edges of the inner radio lobes of
M 84 and 3C 270. Such highly depolarized structures had not been observed in earlier, less detailed
work. The magnetic fields responsible for such depolarization must be tangled on small scales,
since they produce depolarization without any obvious effects on the large-scale Faraday rotation
pattern. The rims are spatially coincident with shells of enhanced X-ray surface brightness, in
which both the field strength and the thermal gas density are likely to be increased by compression.
6.5.3 The radio galaxies 0755+37 and M 87
I have presented two-dimensional analyses of the RM and depolarization across the radio galaxy
0755+37, in a very poor group, and the inner 5 kpc of the radio structure of M 87, located at the
centre of the cool-core Virgo cluster. 0755+37 shows the clearest known example of jet-related
RM structure. The amplitudes of the RM fluctuations across 0755+37 and M 87 are typical of their
environments: in 0755+37 they are smaller than those detected in sources in groups of galaxies and
much more than those in M 87, which are typical of a cool core cluster. The RM structure functions
have been evaluated for both sources over sub-regions where the RM fluctuation amplitudes are
approximately constant. The structure-function analysis has shown that the best-fitting power
spectrum of the RM fluctuations (and thus the magnetic field) is a power law with a cut-off at high
spatial frequencies. The best-fitting power spectra are different for the two lobes of 0755+37: the
slopes are q≃2.5 in the eastern lobe and q≃2.2 in the west. The minimum field scale derived
from depolarization measurements is 0.1 kpc and the maximum field fluctuation scale is≈50 kpc.
127

128 6. Summary and conclusions
In M 87, a power spectrum with slope q≃3.0 fits well everywhere. The minimum scale ranges
from 0.06 up to 0.25 kpc. An approximate lower limit to the outer scale is 2.7 kpc.
6.6 General conclusions
1. Origin of the bulk of the Faraday effect observed across radio galaxies
It is possible to establish that most of the RM is due to material between us and the radio
source by verifying that the polarization angles have aλ 2 dependence and that the detected
depolarization, if any, is consistent with that expected from instrumental effects (beam and
bandwidth depolarization). This is the case for all of the radio galaxies analysed in this
thesis and in most earlier work.
This is consistent with the idea that radio sources have evacuated cavities in the IGM during
their expansion (as observed in X-rays) and that there has been relatively little mixing of
thermal and relativistic plasma. The apparent under-pressure in FR I lobes and tails is most
easily explained if a small amount of thermal plasma has been entrained and heated, but the
densities required are quite small, and consistent with the non-detection of internal Faraday
rotation and depolarization at GHz frequencies.
Prior to the work described here, it was generally thought that the observed RM was
produced almost exclusively by the distributed hot component of the intra-group or intracluster
medium and that interactions with the radio source were unimportant. In the
present study, three distinct types of magnetic-field structure have been found: a component
with large-scale fluctuations, indeed plausibly associated with the undisturbed intergalactic
medium; a well-ordered field draped around the leading edges of radio lobes and a field with
small-scale fluctuations in the shells of compressed gas surrounding the inner lobes, perhaps
associated with a mixing layer.
Foreground Faraday rotation in a spherically symmetrical, undisturbed IGM provides a
natural explanation for the systematic asymmetry in RM variance and/or depolarization
observed between the approaching and receding lobes (Laing 1988; Garrington et al. 1988;
Morganti et al. 1997; Laing et al. 2008). In contrast, some authors (e.g. Bicknell, Cameron
& Gingold 1990) have suggested that the RM could be localized in a thin layer around
the source, but simple shell models of this type find difficulties in reproducing the observed
asymmetries and are unlikely to be important on large scales. It is likely that the sphericallysymmetric
group component dominates in large, tailed sources such as 3C 449.
The results on banded RM structures presented in this thesis suggest that the magnetised
medium producing them must have a scale comparable to that of the lobe width. If the
128

6.6. General conclusions 129
radio source is expanding supersonically, there will also be a bow shock ahead of the lobe
in the hot gas which will naturally produce higher densities and stronger fields in the postshock
material immediately surrounding the lobes. All of the sources which show banded
RM structure show evidence of strong interaction with the surrounding medium, either from
expansion driven by the jets (0206+35, 3C 270, 3C 353) or from X-ray observations (M 84).
The best evidence for mixing between thermal and relativistic plasma comes from the
observation of enhanced depolarization in the inner lobes of M 84, 3C 270 and 0755+37.
It is in these regions that models like those of Bicknell, Cameron & Gingold (1990) are
most likely to apply.
2. The connection between magnetic field and thermal gas
The results of this work are consistent with the idea that the RM fluctuation amplitude in
galaxy groups and clusters scales roughly linearly with density, ranging from a few rad m −2
in the sparsest environments (0755+37) through intermediate values of≈30-100 rad m −2 in
groups (0206+35, 3C 270 and 3C 449) to∼ 10 4 rad m −2 in the centres of clusters with cool
cores (M 87).
The spatial variation of RM fluctuation amplitude in galaxy groups is consistent with a
distribution for the magneto-ionic medium which is spherically symmetric, together with
cavities coincident with the radio emission. The radial dependence of the field strength can
be approximated by B(r)=〈B 0 〉 ( )
n e (r) η,
n 0
with 1 > ∼ η> ∼ 0.5. The central magnetic-field
strength B 0 is typically a fewµG and the magnetic field is never dynamically dominant over
the modeled volumes.
Thus the relation between magnetic field strength and density in galaxy groups appears to
be a continuation of the trend observed in galaxy clusters (e.g. Brunetti et al. 2001; Dolag
et al. 2006; Guidetti et al. 2008; Bonafede et al. 2010): the radial variations have similar
functional forms and the normalization scales with density.
3. Magnetic-field structure
In this thesis, two types of RM structures have been found:
one with RM variations
reasonably approximated as isotropic and random (3C 449, M 87, the western lobe of
3C 353) and one with two-dimensional banded RM patterns (0206+35, 3C 270, M 84, the
eastern lobe of 3C 353 and perhaps 0755+37). The first type of structure is well reproduced
by a Gaussian isotropic random magnetic field, as shown by earlier numerical modelling as
well as that performed in this thesis. The second type shows evidence for a two-dimensional,
ordered magnetic field. One of the major results of this thesis has been to demonstrate
that RM enhancements are expected from compression of the surrounding magneto-ionic
medium by expanding radio galaxies, but that such structures can have band-like shapes
129

130 6. Summary and conclusions
with the observed properties only when the field is wrapped around the radio lobes in a twodimensional
geometry. In order to create RM bands with multiple reversals, more complex
field geometries such as two-dimensional eddies are needed.
4. Magnetic field power spectrum: functional form and range of scales
Recent observational work on galaxy clusters has led to claims that the intra-cluster
magnetic field has a Kolmogorov-like power spectrum (i.e. a power law with slope q =
11/3). Kolmogorov turbulence would be expected over some inertial range if the turbulence
is primarily hydrodynamic (Kolmogorov 1941) or for the MHD cascade investigated
by Goldreich & Sridhar (1997). However, the assumptions on which the Kolmogorov
turbulence relies (homogeneity, isotropic and lack of magnetization) are not satisfied in the
intergalactic medium. Numerical simulations predict a wide range of spectral slopes (albeit
with some preference for the Kolmogorov value; Dolag 2006), and a comparison between
theory and observation therefore seems premature.
None of the RM distributions studied in this thesis show fluctuations compatible with a
Kolmogorov power spectrum over the full range of observed scales. Instead, they are
consistent with flatter power-law power spectra (q≤3). A Kolmogorov form on scales
below the resolution limit cannot be excluded, although this is not the case on linear scales
as low as 60 pc in M 87. Power spectra with slopes flatter than 11/3 have also been found
by other authors (Murgia et al. 2004; Govoni et al. 2006; Laing et al. 2008). Similar results
come from the analysis of magnetic turbulence in the Galaxy. For example, Regis (2011)
found a flatter index (q=2.7) by analysing the fluctuations in the Galactic synchrotron
emission (see also Minter & Spangler 1996).
It is worth noting that the power spectra for sources in this thesis appear to be steeper in
richer environments. The slopes range from q≈2 in groups (0755+37, 0206+35) up to
q≈3 in clusters (3C 353, M 87). The flatter slopes are found in sources showing anisotropic
RM’s, but this might be a selection effect. The idea of that the index of the power spectrum
is determined by the richness of the environment is probably oversimplified.
In 3C 449, the power spectrum cannot be a single power law, but rather steepens on smaller
scales. A broken power law gives a good representation of the data, as found for 3C 31
(Laing et al. 2008), but is not unique: a smoother function would fit just as well. The
precise location of the break in the broken power law may not be physically significant, but
a change in slope might occur at the typical field reversal scale if the field is produced by a
fluctuation dynamo (Schekochihin & Cowley 2006; Eilek & Owen 2002).
It is not clear what determines the minimum and the maximum scales of the magnetic field.
Reconnection of field lines is expected to occur on much smaller scales than those sampled
130

6.7. Future prospects 131
and is therefore not relevant. Energy can be injected on large scales from the interactions
between the radio source and the magnetized gas, gas motions (“sloshing”) within the
group/cluster potential well (e.g. merger-related) and motion of the host galaxy. All of
these provide energy input on roughly the outer scale estimated for 3C 449 and 0755+37
(the latter subject to confirmation by three-dimensional modelling). In principle, the origin
of the maximum scale could be constrained by observations of the same type of radio source
in different environments.
5. RM structure and its connection with the radio source morphology
I pointed out a potential correlation between radio source morphology and RM anisotropy.
RM bands have so far been unambiguously detected only in lobed radio galaxies. This is
what expected if the process producing bands is related to some form of compression or
draping. Radio galaxies with lobes are indeed expected to be more effective than tailed
sources in compressing the surrounding gas. Production of RM bands by draped fields in
tailed sources is not excluded, provided that there is relative motion between the source and
the surrounding gas. This might occur in narrow-angle tail sources (where the host galaxy
is moving rapidly through the ICM) or in wide-angle tails if there are sloshing motions of
gas in the cluster potential well.
6.7 Future prospects
The work described in this thesis raises a number of observational questions, including the
following.
1. How common are anisotropic RM structures? Do they occur primarily in lobed radio
galaxies with small axial ratios, consistent with jet-driven expansion into an unusually dense
surrounding medium? Is their frequency qualitatively consistent with the two-dimensional
draped-field picture?
2. Why do we see bands primarily in sources where the isotropic RM component has a flat
power spectrum of low amplitude? Is this just because the bands can be obscured by largescale
fluctuations, or is there a causal connection?
3. Are the RM bands suggested in tailed sources such as 3C 465 and Hydra A caused by a
similar phenomenon (e.g. bulk flow of the IGM around the tails)?
4. Is an asymmetry between approaching and receding lobes seen in the banded RM
component? If so, what does that imply about the field structure?
131

132 6. Summary and conclusions
5. How common are the regions of enhanced depolarization at the edges of radio lobes? How
strong is the field and what is its structure? Is there evidence for the presence of a mixing
layer?
It should be possible to address all of these questions using a combination of observations
with the new generation of synthesis arrays (EVLA, e-MERLIN and LOFAR all have wide-band
polarimetric capabilities) and high-resolution X-ray imaging. It will be possible to determine
rotation measures within a single observing band at high spatial resolution, allowing at the same
time a much quicker and more accurate RM determination and the building of larger samples.
Higher resolution and improved image quality will allow sampling of a larger range of spatial
scales and hence improve the constraints on the RM power spectrum. In particular, it should
be possible to determine with certainty whether the power spectrum has the form expected for
Kolmogorov turbulence. An alternative to fitting parametrized functions to the power spectrum is
provided by Bayesian methods, which are non-parametric and should improved error estimates
(Enßlin & Vogt 2005). RM modelling should be conducted in synergy with jet modelling,
which provides constraints on the geometry of the sources and estimates of their jet powers, and
with X-ray observations. Finally more detailed MHD simulations of radio-source evolution in a
surrounding magnetised medium are needed.
132

Appendix
133

Appendix
Data reduction and imaging of the radio
galaxies 0206+35, 0755+37 and M 84 ‡
The aim of this appendix is to present the VLA data reduction and high-quality radio
imaging of the FR I sources 0206+35, 0755+37 and M 84, which have been performed
in collaboration with R. A. Laing, A. H Bridle, P.Parma and M.Bondi.
Here, we describe:
1. details of the observations and data reduction
2. the source morphologies in total intensity at a range of resolutions,
3. images of the degree of polarization and the apparent magnetic field direction, corrected for
Faraday rotation.
.1 Observations and VLA data reduction
We made use of new and archive VLA observations. Deep images with angular resolution
ranging from 4.5 down to 0.35 arcsec FWHM have been produced from multi-configuration VLA
observations at 1.4 GHz (L-band) and 4.9 GHz (C-band). A journal of the observations is given in
Table 5.
The VLA data listed in Table 5 were calibrated and imaged using the AIPS software package,
following standard procedures, with a few additions. The flux-density scale was set using
observations of 3C 286 or 3C 48 and the zero-point of E-vector position angle was determined
‡ part of Laing, Guidetti et al. 2011, in prep.
135

136 Data reduction and imaging of lobed radio galaxies
Table 5: Journal of VLA observations. ν and∆ν are the centre frequencies and bandwidth,
respectively, for the one or two frequency channels observed and t is the on-source integration
time scaled to an array with all 27 antennas operational.
Source Config. Date ν ∆ν t Proposal
[GHz] [MHz] [min] code
0206+35 A 2008 Oct 13 4885.1, 4835.1 50 486 AL797
A 2008 Oct 18 4885.1, 4835.1 50 401 AL797
B 2003 Nov 17 4885.1, 4835.1 50 254 AL604
C 2004 Mar 20 4885.1, 4835.1 50 88 AL604
A 2004 Oct 24 1385.1, 1464.9 25 189 AL604
B 2003 Nov 17 1385.1, 1464.9 25 110 AL604
0755+37 A 2008 Oct 05 4885.1, 4835.1 50 477 AL797
A 2008 Oct 06 4885.1, 4835.1 50 383 AL797
B 2003 Nov 15 4885.1, 4835.1 50 332 AL604
B 2003 Nov 30 4885.1, 4835.1 50 169 AL604
C 2004 Mar 20 4885.1, 4835.1 50 125 AL604
D 1992 Aug 02 4885.1, 4835.1 50 55 AM364
A 2004 Oct 25 1385.1, 1464.9 12.5 450 AL604
B 2003 Nov 30 1385.1, 1464.9 12.5 160 AL604
C 2004 Mar 20 1385.1, 1464.9 12.5 21 AL604
M 84 A 1980 Nov 09 4885.1 50 223 AL020
A 1988 Nov 23 4885.1, 4835.1 50 405 AW228
A 2000 Nov 18 4885.1, 4835.1 50 565 AW530
B 1981 Jun 25 4885.1 50 156 AL020
C 1981 Nov 17 4885.1 50 286 AL020
C 2000 Jun 04 4885.1, 4835.1 50 138 AW530
A 1980 Nov 09 1413.0 25 86 AL020
B 1981 Jun 25 1413.0 25 29 AL020
B 2000 Feb 09 1385.1, 1464.9 50 30 AR402
using 3C 286 or 3C 138, after calibration of the instrumental leakage terms. The main deviations
from standard methods were as follows.
Firstly, we used the routineBLCAL to compute closure corrections for the 4.9 GHz observations.
This was required to correct for large closure errors on baselines between EVLA and VLA
antennas in the most recent observations (VLA Staff 2010), but also improved a number of the
earlier datasets. Whenever possible, we included observations of the bright, unresolved calibrator
3C 84 for this purpose; if it was not accessible during a particular run, we used 3C 286. We found
that it was not adequate to use the standard calibration (which averages over scans) to compute the
baseline corrections, as phase jumps during a calibrator scan caused serious errors in the derived
corrections. We therefore self-calibrated the observations in amplitude and phase with a solution
interval of 10 s before runningBLCAL. We assumed a point-source model for 3C 84 and the well-
136

.1. Observations and VLA data reduction 137
determinedCLEAN model supplied with the AIPS distribution for 3C 286.
Secondly, we imaged in multiple facets to cover the inner part of the primary beam at L-band
and to image confusing sources at large distances from the phase centre in both bands. Before
combining configurations, we subtracted in the (u, v) plane all sources outside a fixed central field.
For 0755+37, this procedure failed to remove sidelobes at the centre of the field from a bright
confusing source close to the half-power point of the primary beam. The reason for this is that the
VLA primary beam is not azimuthally symmetric, so the effective complex gain for a distant source
is not the same as that at the pointing centre and varies with time in a different way. We used the
AIPS procedurePEELR to remove the offending source from the (u, v) data for each configuration
before combining them.
Finally, we corrected for variations in core flux density and amplitude scale between
observations as described in Laing et al. (2006b).
J2000 coordinates are used throughout this work. The astrometry for each of the sources was
set using the A-configuration observations, referenced to a nearby phase calibrator in the usual
manner. Thereafter, the position of the compact core was held constant during the process of array
combination. If positions from archival data were originally in the B1950 system, then (u,v,w)
coordinates were recalculated for J2000 before imaging. The C-band data were usually taken in
two adjacent 50 MHz frequency channels, which were imaged together. We also made I images at
L-band using the data from both channels; these were used for spectral-index analysis which will
be presented elsewhere.
In order to avoid the well-known problems introduced by the conventionalCLEAN algorithm
for well-resolved, diffuse brightness distributions, total-intensity images at the higher resolutions
were produced using either a maximum-entropy algorithm (Leahy & Perley 1991) or the multiscaleCLEAN
algorithm as implemented in the AIPS package (Greisen et al. 2009). In the former
case, bright, compact core and jet components were firstCLEANed out. The maximum-entropy
images were then convolved with a Gaussian beam and theCLEAN components restored. We found
very few differences between the images produced by the two methods. The standard singleresolutionCLEAN
was found to be adequate for the lowest-resolution I images. Stokes Q and U
images wereCLEANed using one or more resolutions (we found few differences between single and
multiple-resolutionCLEAN, for these images, which have little power on large spatial scales). All
of the images were corrected for the effects of the antenna primary beam.
In general, the 4.9 GHz images have off-source rms levels very close to those expected from
thermal noise in the receivers alone. The integrations for the L-band images are shorter than at the
higher frequency, and the noise levels are correspondingly higher. As a check on the amplitude
calibration and imaging of the I images used in spectral-index analysis, we have integrated the
137

138 Data reduction and imaging of lobed radio galaxies
Table 6: Resolutions and noise levels for the images used in this work. Col.1 source name; Col.2
angular resolution; Col.3 observing frequency; Col.4 VLA configurations; Col.5 imaging method;
Col.6 off-source noise level on the I image (σ I ) and the average of the noise levels for Q and
U (σ P ). Col.7 approximate maximum scale of structure imaged reliably (Ulvestad, Perley &
Chandler 2009). Col.8 corrected single-dish flux densities put in the standard scale of Baars et al.
(1977), as used at the VLA. Col.9 references.
Source FWHM Freq Config. Method rms noise level Max I int /Jy Ref.
[arcsec] [GHz] I QU [µJy beam −1 ] scale Image SD
σ I σ P [arcsec]
0206+35 1.20 4860.1 BC MR MR 12 12 300 0.90±0.02 0.98±0.12 1
1.20 1464.9 AB MR MR 21 22 300 2.12±0.04 2.13 3
1.20 1385.1 AB MR MR 21 23 300 2.12±0.04 2.22 3
1.20 1425.0 AB MR − 19 − 300 2.12±0.04 2.18 4
4.50 4860.1 BC MR − 18 − 300 0.90±0.02 0.98±0.12 1
4.50 1425.0 AB MR − 38 − 300 2.13±0.04 2.18 4
0755+37 1.30 4860.1 BCD MR MR 7.8 7.9 300 1.26±0.03 1.27±0.02 2
1.30 1464.9 ABC MR MR 28 28 300 2.60±0.05 2.53±0.09 2
1.30 1385.1 ABC MR MR 27 26 300 2.74±0.05 2.62±0.09 2
1.30 1425.0 ABC MR − 20 − 300 2.64±0.05 2.57±0.09 2
4.00 4860.1 BCD MR MR 14 12 300 1.25±0.03 1.27±0.02 2
4.00 1464.9 ABC MR MR 44 42 300 2.59±0.05 2.53±0.09 2
4.00 1385.1 ABC MR MR 46 36 300 2.73±0.05 2.62±0.09 2
4.00 1425.0 ABC MR MR 32 − 300 2.65±0.05 2.57±0.09 2
M 84 1.65 4860.1 a ABC MR MR 15 11 300 2.94±0.06 2.88±0.08 3
1.65 1413.0 AB MR MR 140 140 120 6.03±0.12 6.44±0.24 3
4.5 4860.1 a C MR MR 23 20 300 2.98±0.06 2.88±0.08 3
4.5 1464.9 B MR MR 120 70 120 6.14±0.12 6.32±0.24 3
4.5 1413.0 AB MR MR 210 150 120 5.99±0.12 6.44±0.24 3
4.5 1385.1 B MR MR 130 69 120 6.46±0.13 6.51±0.24 3
References: 1 Gregory & Condon (1991); 2 Kühr et al. (1981); 3 Laing & Peacock (1981); 4 White & Becker (1992).
a Although the 1980 and 1981 observations have a centre frequency of 4885.1 MHz, the weighted mean for the combined dataset is
still close to 4860.1 MHz.
flux densities using the AIPS verbTVSTAT. We estimate that the errors are dominated by a residual
scale error of≈2%. All of the results are in excellent agreement with single-dish measurements
(Table 6).
The configurations, resolutions, deconvolution algorithms and noise levels for the final images
are listed in Table 6. The noise levels were measured before correction for the primary beam, and
are appropriate for the centre of the field.
.2 Images
Our conventions are as follows:
1. images of total intensity are show as grey-scales, over a range indicated by the labelled
wedges.
138

.2. Images 139
2 4 6
2 4 6 8 10
35 48 30
(g)
15
DECLINATION (J2000)
00
47 45
30
15
00
02 09 43 42 41 40 39 38 37 36 35 34
RIGHT ASCENSION (J2000)
p = 1
02 09 43 42 41 40 39 38 37 36 35 34
RIGHT ASCENSION (J2000)
Figure 11: 0206+35 at 4.9 GHz with angular resolution of 1.2 arcsec FWHM. Left: total intensity
distribution I in mJy (beam area) −1 . Right: apparent B a vectors with lengths proportional to p 4.9
derived from the 3-frequency RM fit, superimposed on a false-colour display of I.
2. We also show intensity gradient for 0755+37 and M 84,|∇I|, derived using a Sobel filter
(Sobel & Feldman, 1968).
3. Linear polarization is illustrated by plots in which vectors have lengths proportional to
the degree of polarization at 4.9 GHz (p 4.9 ) and directions along the apparent magnetic
field (B a ). They are superposed on false-colour images of either I (again with a labelled
wedge indicating the range) or|∇I|. B a ) vectors are plotted where I ≥ 5σ I . A value of
p = 1 is indicated by the labelled bar. The apparent field direction isχ 0 +π/2, where
χ 0 is the E-vector position angle corrected to zero-wavelength by fitting to the relation
χ(λ 2 )=χ 0 + RMλ 2 for foreground Faraday rotation. In some sources, we used the RM
images at lower resolution to correct the position angles. This procedure is valid if the RM
varies smoothly over the low-resolution image, and maximises the area over which we can
determine the direction of the apparent field.
4. The restoring beam is shown in the bottom left-hand corner of each plot.
5. We refer to parts of the sources by the abbreviations N, S, E, W (for North, South, East,
West) etc.
6. We refer to the main (brighter) and counter (fainter) jets.
139

140 Data reduction and imaging of lobed radio galaxies
.2.1 0206+35
The left panel of Fig. 11 shows the total intensity distribution over all of 0206+35 at 4.9 GHz,
1.2 arcsec resolution.
At this resolution, both lobes are circular in cross-section and show sharp outer boundaries,
particularly to the NW and SE of the source, plus some fainter diffuse emission to the N and S.
If the orientation ofθ≈40 ◦ determined for the inner jets (Laing & Bridle, in preparation) also
applies to the lobes, then they are presumably ellipsoidal with an axial ratio≈1. The figure also
shows internal structure in both jets. NW jet has the brighter base, and both bends and brightens
as it enters its lobe, after which its path meanders. The SE counter-jet appears to expand more
rapidly initially, then also meanders as it enters its lobe. The right panel of Fig. 11 illustrates that
the magnetic field configuration in both lobes is well ordered and basically circumferential, while
near the central lines of the jets it is predominantly perpendicular to their axis, with evidence for
parallel field at the jet edges. The southern edge of the source is strongly polarized with magnetic
field tangential to the source boundary.
.2.2 0755+37
Figs. 12(a) and (b) show the total intensity distribution over all of 0755+37 at two resolutions
and frequencies. Fig. 12(a) at 1.4 GHz, 4.0 arcsec resolution, shows that the large scale structure
consists of two lobes, again roughly circular in projection, with well-defined but not particularly
sharp outer edges to the W and E, plus fainter diffuse emission to the N and S. The E lobe has a
series of narrow rings and brightness steps in the region where the brighter jet appears to terminate.
They are recessed from the eastern boundary of this lobe and some may be the edges of thin shells.
The structure of the W lobe is unusual, containing some arc-like features. and other structure
suggestive of a rapidly decollimating counter-jet W of the nucleus, as previously described by
Bondi et al. (2000), with a “hole”, or deficit of emission in the region where the counter-jet
might be expected to terminate. Although partially resolved out, all of these internal structures
are confirmed by the higher resolution image at 4.9 GHz, 1.3-arcsec resolution (Fig. 12(b)).
Figs. 12(c) and (d) show that the magnetic field in both lobes is exceptionally well organised, and
mainly tangential to the lobe boundaries, at both the angular resolutions. At the lower resolution,
the degree of linear polarization p≤0.6 over much of both lobes. Noteworthy is the excellent
alignment between the field vectors and the ridges of high brightness gradient on both sides of the
source.
140

.2. Images 141
0 2 4 6 8 10 12 14
37 48 30
15
00
(a)
(c)
DECLINATION (J2000)
47 45
30
15
00
46 45
30
15
00
0.0 0.5 1.0 1.5 2.0 2.5
p = 1
0.5 1.0 1.5 2.0 2.5
37 48 30
15
00
(b)
(d)
DECLINATION (J2000)
47 45
30
15
00
46 45
30
15
00
07 58 36 34 32 30 28 26 24 22 20
RIGHT ASCENSION (J2000)
p = 1
07 58 36 34 32 30 28 26 24 22 20
RIGHT ASCENSION (J2000)
Figure 12: images of 0755+37 at 4.0 arcsec (panels a and c) and 1.3 arcsec (panels b and d). (a)
Total intensity at 1.4 GHz in mJy beam −1 . (b) Total intensity at 4.9 GHz in mJy beam −1 . Apparent
B a vectors with lengths proportional to p 4.9 from the 3-frequency RM fits, superimposed on a
false-colour display at 4.9 GHz of the intensity gradient (c) and the total intensity (d). The panels
show identical areas.
.2.3 M 84
The left panel of Fig. 13 shows the total intensity distribution over M 84 at 1.65 arcsec FWHM
resolution. The source appears to be an intermediate case between lobed and tailed sources: the
lobes, in particular the NE one, more closely resemble the inner plumes in FR I sources(3C 31,
Laing et al. 2008), albeit on a much smaller linear scale (∼6 kpc) and somewhat rounder shape.
On the other hand, the spectral gradients (Laing et al. in preparation) are more characteristic of
lobed sources, with no hint of steepening outwards. Both jets expand rapidly and deflect within
about 1 arcmin of the nucleus. They are surrounded by diffuse emission, at least in projection,
everywhere except perhaps within a few arcsec of the nucleus. The North jet is brighter at its base
and forms a plume-like structure without well-defined edges after deflecting though approximately
141

142 Data reduction and imaging of lobed radio galaxies
0 2 4 6 8 10
12 54 30
00
53 30
DECLINATION (J2000)
00
52 30
00
51 30
12 25 09 08 07 06 05 04 03 02 01
RIGHT ASCENSION (J2000)
p = 1
12 25 09 08 07 06 05 04 03 02 01
RIGHT ASCENSION (J2000)
Figure 13: Images of M 84 at 1.65 arcsec resolution. Left: Total intensity at 4.9 GHz in mJy
(beam area) −1 . Right: Apparent B a vectors with lengths proportional to p 4.9 derived from the
4-frequency RM fits at 4.5 arcsec, superimposed on a false-colour display of the total intensity
gradient at 4.9 GHz.
90 ◦ in projection. Lower-resolution observations were previously shown by Laing & Bridle (1987)
so are not repeated here.
The right panel of Fig. 13 shows the apparent magnetic field structure. In the N lobe, the
magnetic field is predominantly perpendicular to the presumed path of the jet along its mid-line.
The magnetic field in the S lobe is broadly circumferential and a sudden increase in the degree of
polarization at the edge of the lobe is apparent.
142

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Acknowledgments
Probably the most important result is that during this PhD I have grown not only as scientist but
as person. For this, I owe many people a debt of gratitude for all my life.
I am privileged for having Robert Laing as supervisor, this thesis would not have been possible
without his help and patience. I have learned many valuable things from him, and I also wish to
express gratitude for believing in me. One simply could not wish for a better supervisor.
I thank Paola Parma and Loretta Gregorini for their help and support.
I would like to acknowledge ESO (Garching) for the award of a ESO Studentship that allowed
me to breath a wide spirit of research and to grow.
I wish to thank Matteo Murgia, Federica Govoni and the staff of the Astronomical Observatory
of Cagliari for their hospitality and support during the development of the work on 3C 449.
The work on this radio galaxy is part of the “Cybersar” Project, which is managed by the
COSMOLAB Regional Consortium with the financial support of the Italian Ministry of University
and Research (MUR), in the context of the “Piano Operativo Nazionale Ricerca Scientifica,
Sviluppo Tecnologico, Alta Formazione (PON 2000-2006)”. I wish to thank Luigina Feretti for
providing the excellent VLA dataset of 3C 449 and Greg Taylor for the use of his rotation measure
code. I also thank M. Swain for providing the VLA dataset for 3C 353, J. Eilek for a FITS image
of 3C 465 and J. Croston, A. Finoguenov and J. Goodger for the FITS images of 3C 270, M 84 and
3C 353, respectively.
I am grateful to M. Bondi, G. Brunetti, A. Shukurov and J. Stöckl for many valuable
comments. I also acknowledge the use of HEALPIX package (http://healpix.jpl.nasa.gov) and
the provision of the models of Dineen & Coles (2005) inHEALPIX format.
The last but certainly not least words are for all of You which “see all my light and love my
dark”.
“And thus I stand”.
153